The Gabrielov–Khovanskii Problem for Polynomials
Liverpool L69 7ZL UK
pukh@liv.ac.uk
Abstract
We state and consider the Gabrielov–Khovanskii problem of estimating the multiplicity of a common zero for a tuple of polynomials in a subvariety of a given codimension in the space of tuples of polynomials. For a bounded codimension we obtain estimates of the multiplicity of the common zero, which are close to optimal ones. We consider certain generalizations and open questions.
Keywords
1 Introduction
1.1 The Gabrielov–Khovanskii Problem
Let ${\mathbb{C}}^{N}$ be the complex coordinate space and ${\cal F}_{1},\ldots,{\cal F}_{N}$ some linear spaces of functions, analytic in a neighborhood of the point $o=(0,\ldots,0)$ and vanishing at that point. For an arbitrary tuple $(f_{1},\ldots,f_{N})\in{\cal F}_{1}\times\cdots\times{\cal F}_{N}$ we define the multiplicity of zero $\mu(f_{1},\ldots,f_{N})\in{\mathbb{Z}}_{+}\cup\infty$:

•
if the set $\{f_{1}=\cdots=f_{N}=0\}$ has a component of positive dimension, passing through the point $o$, we set $\mu(f_{1},\ldots,f_{N})=\infty$,

•
otherwise, $\mu(f_{1},\ldots,f_{N})$ is the multiplicity of the isolated common zero $o$ of the functions $f_{1},\ldots,f_{N}$, that is, the integer ${\rm dim}_{\mathbb{C}}{\cal O}/(f_{1},\ldots,f_{N})$.
By the Gabrielov–Khovanskii problem we mean the following question: what is the codimension of the closed subset
$\displaystyle{\cal F}(m)=\{(f_{1},\ldots,f_{N})\,\,\mu(f_{1},\ldots,f_{N}) \geqslant m\}$ 
in the space ${\cal F}={\cal F}_{1}\times\cdots\times{\cal F}_{N}$?
There is an obvious dual form of this problem. Instead of fixing a multiplicity $m\in{\mathbb{Z}}_{+}\cup\{\infty\}$ and computing or estimating the codimension of the set ${\cal F}(m)$, one can fix (or bound from above) the codimension of an irreducible closed subset in ${\cal F}$ and compute or estimate the multiplicity of a generic tuple in that subset. More precisely, let $B\subset{\cal F}$ be an irreducible closed subset of codimension $a\in{}\mathbb{Z}_{+}$. Set
$\displaystyle\mu(B)={\rm min}\{\mu(\underline{f})\,\,\underline{f}=(f_{1}, \ldots,f_{N})\in B\},$ 
that is, $\mu(B)$ is the multiplicity at zero of the general tuple of functions $\underline{f}\in B$. Now the Gabrielov–Khovanskii problem takes the form of the question: what is the maximal multiplicity
$\displaystyle\mu(a)={\rm max}\{\mu(B)\,\,{\rm codim}(B\subset{\cal F}) \leqslant a\}?$ 
Obviously, if $\mu(a)=m$, then for the codimension of the set ${\cal F}(m)$ we get the estimate
$\displaystyle{\rm codim}({\cal F}(m)\subset{\cal F})\leqslant a,$ 
and if, conversely, the codimension of the set ${\cal F}(m)$ in ${\cal F}$ is some $a^{*}\in{\mathbb{Z}}_{+}$, then for an irreducible component $B^{*}\subset{\cal F}(m)$ we get $\mu(B^{*})\geqslant m$, so that $\mu(a^{*})\geqslant m$. This is what we mean by the duality of the two forms of the Gabrielov–Khovanskii problem.
Thus we can either fix the multiplicity and look for (or estimate) the codimension, or fix the codimension and estimate the multiplicity. The second form is more natural from the viewpoint of certain geometric applications (see Sect. 1.3), and in this paper it is the second form that we consider. [For the original and most general form of the Gabrielov–Khovanskii problem see their original paper ([Gabrielov and Khovanskii1998])]. For the spaces ${\cal F}_{i}$ we take the spaces of polynomials of degree $d_{i}\geqslant 2$, vanishing at the point $o$; some natural generalizations of the Gabrielov–Khovanskii problem are stated below in Sect. 1.2.
Example 1.1.
Let us compute $\mu(1)$ for $N=2$. Let $B\subset{\cal F}_{1}\times{\cal F}_{2}$ be an irreducible hypersurface. For a general tuple of polynomials $(f_{1},f_{2})\in B$ the curves $C_{1}=\{f_{1}=0\}$ and $C_{2}=\{f_{2}=0\}$ are nonsingular at the point $o$ (otherwise, ${\rm codim}(B\subset{\cal F})\geqslant 2)$. If the tangents $L_{i}=T_{o}C_{i}$ are distinct, then $\mu(f_{1},f_{2})=1$. Otherwise, the tangents coincide for a general tuple $(f_{1},f_{2})\in B$, therefore
$\displaystyle B\subset\{(f_{1},f_{2})\,\,df_{1}(o)\df_{2}(o)\}.$ 
However, the latter set is closed, irreducible and of codimension 1 in ${\cal F}_{1}\times{\cal F}_{2}$, so that the inclusion sign can be replaced by the equality. But then for a general tuple $(f_{1},f_{2})\in B$ the curves $C_{1}$ and $C_{2}$ have simple tangency at the point $o$, so that $\mu(B)=2$ and for that reason $\mu(1)=2$.
Example 1.2.
Let us compute $\mu(2)$ for $N=2$. In the notations of the previous example set:
$\displaystyle B_{i}=\left\{(f_{1},f_{2})\,\left\,\frac{\partial f_{i}}{ \partial z_{1}}(o)=\frac{\partial f_{i}}{\partial z_{2}}(o)=0\right.\right\}.$ 
Obviously, $B_{i}$ is an irreducible closed subset of codimension 2. For a general tuple $(f_{1},f_{2})\in B_{1}$ the curve $C_{1}$ has multiplicity 2 at the point $o$, whereas the curve $C_{2}$ is nonsingular at the point $o$, and moreover the tangent line $L_{2}$ is not tangent to $C_{1}$ at the point $o$, so that $\mu(B_{1})=2$ (and $\mu(B_{2})=2$). Therefore, if $B\subset{\cal F}_{1}\times{\cal F}_{2}$ is a closed irreducible subset of codimension 2, different from $B_{1}$ and $B_{2}$, then for a general tuple $(f_{1},f_{2})\in B$ the curves $C_{1},C_{2}$ are nonsingular at the point $o$. Set $B^{\circ}_{3}\subset{\cal F}_{1}\times{\cal F}_{2}$ to be the set of such tuples $(f_{1},f_{2})$, that the curves $C_{1},C_{2}$ are nonsingular at the point $o$, and moreover
$\displaystyle{\rm ord}_{o}f_{2}_{C_{1}}\geqslant 3.$ 
It is easy to see that the closure $B_{3}=\overline{B^{\circ}_{3}}$ is irreducible, of codimension 2 in ${\cal F}_{1}\times{\cal F}_{2}$ and moreover for a general tuple ($f_{1},f_{2})\in B_{3}$ the equality ${\rm ord}_{o}f_{2}_{C_{1}}=3$ holds. This implies that $\mu(B_{3})=3$ and for any irreducible subset $B\subset{\cal F}_{1}\times{\cal F}_{2}$ of codimension 2, which is not $B_{1}$, $B_{2}$ or $B_{3}$, we have $\mu(B)\leqslant 2$. Therefore, $\mu(2)=3$.
Example 1.3.
The computations of Example 1.1 generalize easily for an arbitrary number of variables $N$. The closed subset
$\displaystyle B^{*}=\{(f_{1},\ldots,f_{N})\,\,{\rm rk}(df_{1}(o),\ldots,df_{N }(o))=N1\}$ 
is of codimension 1 in ${\cal F}_{1}\times\cdots\times{\cal F}_{N}$. Obviously, $\mu(B^{*})=2$ and for any irreducible hypersurface $B\neq B^{*}$ we have $\mu(B)=1$. Therefore, $\mu(1)=2$.
It is clear that $\mu(a)<\infty$ if and only if
$\displaystyle a<{\rm codim}({\cal F}(\infty)\subset{\cal F}).$ 
(This, however, does not mean that the Gabrielov–Khovanskii problem makes sense only for those values of $a$, see Sect. 1.2).
Example 1.4.
Let $N=2$ and $d_{1}\leqslant d_{2}$. The closed set ${\cal F}(\infty)$ is reducible: it consists of such tuples $(f_{1},f_{2})$, that either one of the polynomials $f_{i}$ is identically zero or (in the notations of Example 1.1) the curves $C_{1},C_{2}$ have a common component. The degree of that component parametrizes irreducible components of the set ${\cal F}(\infty)$. It is easy to check that the least codimension is that of either component consisting of such tuples $(f_{1},f_{2})$, that the curves $C_{1},C_{2}$ have a line $L\ni o$ as a common component, or of the component $\{(f_{1},f_{2})\,\,f_{1}\equiv 0\}$. Therefore,
$\displaystyle{\rm codim}({\cal F}(\infty)\subset{\cal F})={\rm min}\left(d_{1} +d_{2}2,\frac{(d_{1}+1)(d_{1}+2)}{2}1\right).$ 
Example 1.5.
Again let $N=2$ and $d_{1}\leqslant d_{2}$. Consider the irreducible subvariety $B\subset{\cal F}$, given by the condition
$\displaystyle B=\{(f_{1},f_{2})\,\,{\rm mult}_{o}C_{i}\geqslant m,\quad i=1,2\},$ 
where $m\leqslant d_{1}$; we use the notations $C_{i}$ introduced in Example 1.1 again. Obviously, ${\rm codim}(B\subset{\cal F})=m^{2}+m2$ and $\mu(B)=m^{2}$. Therefore, the inequality
$\displaystyle\mu(m^{2}+m2)\geqslant m^{2}$ 
holds. The following elementary fact is well known (see, for instance, [Hartshorne1977], ChapterV, Sec.3, Ex.3.2): if the curves $C_{1},C_{2}$ have no common irreducible component passing through the point $o$, then
$\displaystyle\mu(f_{1},f_{2})=\sum_{x\geqslant o}({\rm mult}_{x}C_{1})({\rm mult }_{x}C_{2}),$ 
where the sum is taken over the finite set consisting of the point $o$ and all infinitely near points of intersection of the curves $C_{1}$ and $C_{2}$, lying over the point $o$; the multiplicity of a curve at an infinitely near point $x$ is understood as the multiplicity at $x$ of the strict transform of that curve on the surface where $x$ is a point in the usual sense, that is, the surface obtained by a finite sequence of blow ups. [The set of all points of an algebraic surface and all its infinitely near points, equipped with several natural structures, forms a well known classical object, the “bubble space”; for its detailed description see [Manin Yu1986].] Experimenting with finite sets of infinitely near points, lying over the point $o$ in the same way as it was done in the beginning of this example, we arrive to the following general conjecture.
Conjecture 1.1.

(i)
(Stabilization) For any fixed $N$ there is a function $\delta\colon{\mathbb{Z}}_{+}\to{\mathbb{Z}}_{+}$ such that for ${\rm min}(d_{i})\geqslant\delta(a)$ the number $\mu(a)$ does not depend on the tuple $\underline{d}=(d_{1},\ldots,d_{N})$. Denote it by the symbol $\mu^{\rm st}(a)$.

(ii)
(Asymptotics). There is a finite limit
$\displaystyle\lim\limits_{a\to\infty}\frac{\mu^{\rm st}(a)}{a}.$
Now let us consider the behaviour of the numbers $\mu(a)$ for growing values of the number of variables $N$.
Example 1.6.
(See [Pukhlikov2004], Section 3.5) Let the closed subset $B\subset{\cal F}$ be given by the condition
$\displaystyle{\rm rk}\\partial f_{i}/\partial z_{j}(o)\_{1\leqslant i,j \leqslant N}\leqslant Nb$ 
for $b\in\{1,\ldots,N\}$. It is easy to see that ${\rm codim}(B\subset{\cal F})=b^{2}$. Furthermore, for a general tuple $\underline{f}\in B$ the rank of the Jacobi matrix $\\partial f_{i}/\partial z_{j}(o)\$ equals $Nb$, so that there is a subset $I\subset\{1,\ldots,N\}$, $\sharp I=b$, such that the linear forms $df_{i}(o)$, $i\not\in I$, are linearly independent. Therefore, the subset $X=\{f_{i}=0\,\,i\not\in I\}$ is a smooth subvariety of dimension $b$ around the point $o$. The restriction $f_{i}_{X}$ for $i\in I$ has the zero differential at the point $o$, that is,
$\displaystyle{\rm ord}_{o}(f_{i}_{X})\geqslant 2,$ 
and moreover, for a general tuple $\underline{f}\in B$ we have the equality ${\rm ord}_{o}(f_{i}_{X})=2$. Since no other conditions are imposed on $\underline{f}$, for a general tuples $\underline{f}$ we get:
$\displaystyle\mu(\underline{f})=2^{b}.$ 
Therefore, $\mu(b^{2})\geqslant 2^{b}$. The function $\mu(a)$ is obviously nondecreasing, so that we finally get the inequality $\mu(a)\geqslant 2^{[\sqrt{a}]}$ for $a\leqslant N^{2}$.
Conjecture 1.2.
For $N\geqslant\sqrt{a}$ there is the limit
$\displaystyle\lim\limits_{a\to\infty}\quad\frac{\mu(a)}{2^{\sqrt{a}}}=1$ 
In the present paper we will show a weaker statement: for $a\leqslant N$ the function $\mu(a)$ grows as $C^{\sqrt{a}}$, where $C>0$ is some effectively estimated constant. More precisely (see Theorem 4.3 and Remark 4.3), we obtain an upper bound for $\mu(a)$ which asymptotically behaves as
$\displaystyle\frac{1}{\sqrt{a}}e^{2\sqrt{a}}.$ 
1.2 Open Questions and Generalizations
All the main questions related to the Gabrielov–Khovanskii problem are open. Computing the multiplicities $\mu(a)$ (one should write $\mu(a;\underline{d})$, but the discrete parameter $\underline{d}\in{\mathbb{Z}}_{+}^{N}$ is implicitly meant) seems to be a very difficult problem. It is natural to try to estimate them with various degrees of precision, examples of such estimates are given by Conjectures 1.1 and 1.2.
Here is an example of an open question. As we mentioned in Sect. 1.1, the equality $\mu(a)=\infty$ holds for $a\geqslant{\rm codim}({\cal F}(\infty)\subset{\cal F})$, however this does not mean that the Gabrielov–Khovanskii problem can not be set for such values of $a$. Set
$\displaystyle\mu^{*}(a)={\rm max}\{\mu(B)\,\,B\not\subset{\cal F}(\infty),{ \rm codim}(B\subset{\cal F})\leqslant a\}.$ 
The numbers $\mu^{*}(a)$ are defined for all $a\leqslant{\rm dim}{\cal F}$.
Example 1.7.
Let $N=2$, then
$\displaystyle{\rm dim}{\cal F}=\frac{1}{2}(d^{2}_{1}+3d_{1}+d^{2}_{2}+3d_{2})$ 
and obviously $\mu^{*}({\rm dim}{\cal F})=d_{1}d_{2}$.
For arbitrary $N$ and $a\leqslant{\rm dim}{\cal F}$ computing and estimating the numbers $\mu^{*}(a)$ is a very difficult problem.
Now let us consider a more general setting of the Gabrielov–Khovanskii problem (in the framework of algebraic geometry). Let $X$ be a projective algebraic variety, $o\in X$ some point (not necessarily nonsingular!). Set $N={\rm dim}X$. If the point $o$ is an isolated zero of the system of equations $f_{1}=\cdots=f_{N}=0$, where $f_{i}\in{\cal O}_{o,X}$, $f_{i}(o)=0$, then the Samuel multiplicity
$\displaystyle\mu(\underline{f})=e_{\cal O}/(f_{1},\ldots,f_{N})$ 
where ${\cal O}={\cal O}_{o,X}$, is well defined (when $X$ is nonsingular at the point $o$, this is just the dimension of the quotient algebra ${\cal O}_{o,X}/(f_{1},\ldots,f_{N})$, see Fulton ([Fulton1984], Examples 7.1.2 and 7.1.10). Now let $L_{1},\ldots,L_{N}\in{\rm Pic}X$ be some classes, where $H^{0}(X,L_{i})\neq\{0\}$. Set
$\displaystyle{\cal F}=\{(s_{1},\ldots,s_{N})\,\,s_{i}(o)=0\}\subset\prod^{N}_ {i=1}H^{0}(X,L_{i}).$ 
Locally the sections $s_{i}$ are represented by regular functions $f_{i}\in{\cal O}_{o,X}$, so that the multiplicities $\mu(\underline{s})=\mu(s_{1},\ldots,s_{N})\in{\mathbb{Z}}_{+}\cup\{\infty\}$ are well defined. This makes it possible to define the numbers
$\displaystyle\mu(a)=\mu_{X}(a;L_{1},\ldots,L_{N})$ 
in word for word the same way as it was done in Sect. 1.1 and set the generalized Gabrielov–Khovanskii problem.
Example 1.8.
Let $N=2$, $X={\mathbb{P}}^{1}\times{\mathbb{P}}^{1}$ (the surface $X$ can be naturally seen as a quadric in ${\mathbb{P}}^{3}$), $L_{i}=d_{i}\Delta_{i}$, where $\Delta_{i}$ are the standard generators of the Picard group, ${\rm Pic}X={\mathbb{Z}}\Delta_{1}\oplus{\mathbb{Z}}\Delta_{2}$, that is, $\Delta_{i}$ is the class of a line ${\mathbb{P}}^{1}\times\{\mbox{pt}\}$ or $\{\mbox{pt}\}\times{\mathbb{P}}^{1}$, respectively. Let $o\in X$ be an arbitrary point. It is easy to see that $\mu_{X}(a;L_{1},L_{2})$ is equal to
$\displaystyle{\rm max}\{n_{1}n_{2}\,\,n_{i}\leqslant d_{i},n_{1}+n_{2} \leqslant a+2\}.$ 
For that reason, $\mu_{X}(a;L_{1},L_{2})=(\frac{a}{2}+1)^{2}$, if $a\in 2{\mathbb{Z}}$, and $\frac{(a+1)(a+3)}{4}$, if $a$ is odd.
Finally, one more generalization of the Gabrielov–Khovanskii problem for polynomials will be considered in Sect. 5.
1.3 One Application of the Gabrielov–Khovanskii Problem
Let us describe briefly one important application of the Gabrielov–Khovanskii problem for polynomials. In birational geometry of higherdimensional rationally connected algebraic varieties estimates of the multiplicity of a singular point in terms of the degree of a subvariety are of high importance. Let $X\subset{\mathbb{P}}^{N}$ be an irreducible algebraic variety, $o\in X$ a nonsingular (for simplicity) point. Consider a subvariety $Y\subset X$. One needs to estimate the ratio of the multiplicity to the degree:
$\frac{{\rm mult}_{o}Y}{{\rm deg}Y}\leqslant c,$  (1) 
where the estimate should be true for every subvariety $Y$ of a given codimension (for instance, if $X$ is a sufficiently general hypersurface of degree $N$, where $N\geqslant 4$, then for the codimension ${\rm codim}(Y\subset X)=2$ one can take $c=3/(N1)$, see [Pukhlikov2013b], Chapter 3). Of course, one can always take $c=1$, but this estimate, as a rule, is insufficient, especially in higherdimensional problems. The only efficient method of obtaining such estimates, known today, is the method of hypertangent divisors, the idea of which can be explained by the following example. Let $(z_{1},\ldots,z_{N})$ be a system of affine coordinates with the origin at the point $o$, and
$\displaystyle f=q_{1}+q_{2}+\cdots+q_{k}$ 
a polynomial, such that $f_{X}\equiv 0$, where $q_{i}(z)$ are homogeneous of degree $i$. Now the polynomial
$\displaystyle f_{i}=q_{1}+\cdots+q_{i}$ 
has degree $\leqslant i$, however
$\displaystyle f_{i}_{X}=(q_{i+1}+\cdots+q_{K})_{X},$ 
so that the multiplicity of the divisor $\{f_{i}_{X}=0\}$ at the point $o$ is at least $(i+1)$. If $f_{i}_{Y}\not\equiv 0$, then one can form the effective cycle
$\displaystyle(Y\circ\{f_{i}_{X}=0\})$ 
of codimension ${\rm codim}(Y\subset X)+1$ on $X$, for which the ratio of the multiplicity at the point $o$ to the degree is at least
$\displaystyle\frac{{\rm mult}_{o}Y}{{\rm deg}Y}\cdot\frac{i+1}{i}.$ 
The necessary condition $Y\not\subset\{f_{i}_{X}=0\}$ is provided by the regularity conditions for the equations, defining the variety $X$. For the details and numerous examples of applications of the method of hypertangent divisors to the problems of higherdimensional birational geometry see [Pukhlikov2009]; [Pukhlikov2013b]; [Pukhlikov2015]. The procedure described above is iterated and makes it possible to construct, starting from the subvariety $Y$ (the existence of which is assumed), satisfying the estimate
$\displaystyle\frac{{\rm mult}_{o}Y}{{\rm deg}Y}>c,$ 
an effective 1cycle (that is, an integral linear combination of curves) $C$, such that ${\rm mult}_{o}C>{\rm deg}C$. The latter is impossible, whence we conclude that the inequality (1) holds for all subvarieties $Y\subset X$ of the given codimension.
Unfortunately, for certain classes of Fano varieties the procedure described above gives nothing by itself: applying the techniques of hypertangent divisors, one can construct an effective curve $C$, for which the ratio $({\rm mult}_{o}C/{\rm deg}C)$ is less than 1, although is close to that number. Therefore, no contradiction is obtained and the inequality (1) can not be shown directly. In order to circumvent this obstruction, we use the Gabrielov–Khovanskii problem.
For instance, if $X$ is a Fano complete intersection of quadrics and cubics in ${\mathbb{P}}^{N}$ of index 1 (see [Pukhlikov2013a]), then for any irreducible curve $\Gamma\subset X$ of degree ${\rm deg}\Gamma\geqslant 2$ the estimate
$\displaystyle\frac{{\rm mult}_{o}\Gamma}{{\rm deg}\Gamma}\leqslant\frac{2}{3}$ 
holds, which is sufficient to prove birational rigidity of the variety $X$, provided that the lines passing through the point $o$, form a not too large part of the effective 1cycle $C$, which is the output of the technique of hypertangent divisors. The Gabrielov–Khovanskii problem provides an estimate of the input of the lines. For the details, see [Pukhlikov2013a].
It is this application that initially generated the interest of the author to the Gabrielov–Khovanskii problem, see [Pukhlikov2012].
The author thanks the referee for a number of useful suggestions.
2 Statement of the Problem
In this section we give a precise statement of the Gabrielov–Khovanskii problem for polynomials: we introduce the spaces of tuples of polynomials, biinvariant subvarieties and multiplicities. Then we give an estimate of the codimension of the set of tuples that vanish on a subset of positive dimension. We define the parameter $\beta$, characterizing a subvariety of tuples of polynomials.
2.1 The Space of Tuples of Polynomials
Fix the complex coordinate space ${\mathbb{A}}={\mathbb{C}}^{N}_{(z_{1},\ldots,z_{N})}$ of dimension $N\geqslant 1$ with coordinates $(z_{*})=(z_{1},\ldots,z_{N})$. For $d\in{\mathbb{Z}}_{+}$ let ${\cal P}_{d,N}$ be the linear space of homogeneous polynomials of degree $d$ in $z_{*}$ (in particular, ${\cal P}_{0,N}={\mathbb{C}}$), and for $e\leqslant d$ set
$\displaystyle{\cal P}_{[e,d],N}=\bigoplus\limits^{d}_{i=e}{\cal P}_{i,N},$ 
for instance, ${\cal P}_{[1,d],N}$ is the space of (nonhomogeneous) polynomials of degree $\leqslant d$ with no free term. On each of these spaces acts the matrix group $G_{1}=GL_{N}({\mathbb{C}})$ of linear changes of coordinates. Fix a tuple of integers
$\displaystyle\underline{d}=(d_{1},\ldots,d_{N}),$ 
where $2\leqslant d_{1}\leqslant\cdots\leqslant d_{N}$, and set
$\displaystyle{\cal P}(\underline{d})=\prod^{N}_{i=1}{\cal P}_{[1,d_{i}],N}$ 
to be the space of tuples $(f_{1},\ldots,f_{N})$ of polynomials of degree $\leqslant d_{1},\ldots,d_{N}$, respectively, with no free term. On the space ${\cal P}(\underline{d})$, apart from the abovementioned group $G_{1}$, act two more groups of transformations, which we will now define. The group $G_{21}$ consists of transformations of the form
$\displaystyle(f_{1},\ldots,f_{N})\mapsto(f^{+}_{1},\ldots,f^{+}_{N}),\quad \quad f^{+}_{i}=f_{i}+\sum^{i1}_{j=1}s_{i,j}(z)f_{j},$ 
where $s_{i,j}\in{\cal P}_{[0,d_{i}d_{j}],N}$ are polynomials, fixed for the given transformation. Set ${\cal D}=\{d_{1}\}\cup\cdots\cup\{d_{N}\}\subset{\mathbb{Z}}_{+}$ and let for $d\in{\cal D}$
$\displaystyle n_{d}=\sharp\{i\,\,d_{i}=d\},$ 
so that $\sum_{d\in{\cal D}}n_{d}=N$. Now the group $G_{22}$ is defined as the matrix group (realized by blockwise diagonal matrices, where the blocks of the size $n_{d}\times n_{d}$ are ordered by increasing of the integers $d$)
$\displaystyle\prod_{d\in{\cal D}}GL_{n_{d}}({\mathbb{C}}),$ 
acting on the tuples $(f_{*})\in{\cal P}(\underline{d})$ by linear transformations of the form
$\displaystyle(f_{1},\ldots,f_{N})\mapsto(f_{1},\ldots,f_{N})A.$ 
Let $G_{2}=\langle G_{21},G_{22}\rangle$ be the group of linear transformations of the space ${\cal P}(\underline{d})$, generated by the subgroups $G_{21}$ and $G_{22}$. The group $G_{2}$ is clearly connected, hence irreducible as an algebraic variety. An irreducible subvariety $B\subset{\cal P}(\underline{d})$ (respectively, a map from ${\cal P}(\underline{d})$ to some set) is said to be biinvariant, if it is invariant with respect to the action of both groups $G_{1}$ and $G_{2}$.
2.2 Multiplicities
For a tuple $\underline{f}=(f_{1},\ldots,f_{N})\in{\cal P}(\underline{d})$ we define the multiplicity $\mu(\underline{f})=\mu(f_{1},\ldots,f_{N})\in{\mathbb{Z}}_{+}\cup\infty$, setting:

•
$\mu(\underline{f})=\infty$, if the closed algebraic set $\{f_{1}=\cdots=f_{N}=0\}$ has a component of positive dimension, containing the point $o=(0,\ldots,0)\in{\mathbb{A}}$,

•
$\mu(\underline{f})={\rm dim}{\cal O}_{o,{\mathbb{A}}}/(f_{1},\ldots,f_{N})$, otherwise.
Obviously, the function $\mu\colon{\cal P}(\underline{d})\to{\mathbb{Z}}_{+}\cup\{\infty\}$ is biinvariant. For an arbitrary irreducible subvariety $B\subset{\cal P}(\underline{d})$ set
$\displaystyle\mu(B)={\rm min}_{\underline{f}\in B}\{\mu(\underline{f})\}\in{ \mathbb{Z}}_{+}\cup\infty,$ 
so that $\mu(B)=\mu(\underline{f})$ for a general tuple $\underline{f}\in B$. Furthermore, set
$\displaystyle\mu(a)={\rm max}\{\mu(B)\,\,{\rm codim}(B\subset{\cal P}( \underline{d}))\leqslant a\},$ 
that is, the maximum is taken over all irreducible subvarieties of codimension $a$. If $\langle B\rangle$ is the biinvariant span of the subvariety $B$, that is, the smallest biinvariant subvariety in ${\cal P}(\underline{d})$, containing $B$, then, obviously, $\mu(\langle B\rangle)=\mu(B)$, and moreover ${\rm codim}(\langle B\rangle\subset{\cal P}(\underline{d}))\leqslant({\rm codim }(B\subset{\cal P}(\underline{d}))$, so that the number $\mu(a)$ can be defined as the maximum of the numbers $\mu(B)$ over all biinvariant irreducible subvarieties $B\subset{\cal P}(\underline{d})$ of codimension at most $a$. This obvious remark will be used in the sequel without special references.
Consider the closed subset
$\displaystyle X_{\infty}=\{\underline{f}\in{\cal P}(\underline{d})\,\,\mu( \underline{f})=\infty\}$ 
and set $\chi_{\infty}(\underline{d})={\rm codim}(X_{\infty}\subset{\cal P}(\underline{ d}))$. Obviously, $\mu(B)=\infty$, if and only if $B\subset X_{\infty}$, and $\mu(a)=\infty$ if and only if $a\geqslant\chi_{\infty}(\underline{d})$. Consider the irreducible subvariety $X_{\rm line}\subset{\cal P}(\underline{d})$, consisting of such tuples $\underline{f}$, that
$f_{1}_{L}\equiv\cdots\equiv f_{N}_{L}\equiv 0$  (2) 
for some line $L\ni o$.
Proposition 2.1.
The following equality holds:
$\displaystyle{\rm codim}(X_{\rm line}\subset{\cal P}(\underline{d}))=1N+\sum^ {N}_{i=1}d_{i}.$ 
Proof.
a trivial dimension count. When a line $L\ni o$ is fixed, the condition (2) defines an irreducible subvariety (in fact, a linear subspace) of codimension $\sum^{N}_{i=1}d_{i}$. Since a general tuple $\underline{f}\in X_{\rm line}$ vanishes on exactly one line, considering the direct product ${\mathbb{P}}^{N1}\times{\cal P}(\underline{d})$ and the second projection, we complete the proof in the standard way. $\square$
Set $X^{+}_{\infty}=\overline{X_{\infty}\backslash X_{\rm line}}$.
Proposition 2.2.
The following estimate holds:
$\displaystyle{\rm codim}(X^{+}_{\infty}\subset{\cal P}(\underline{d})) \geqslant d_{1}N+1.$ 
Proof.
We use the technique developed in Pukhlikov ([Pukhlikov2001], Section 3). The space ${\mathbb{A}}$ is considered as embedded in the projective space ${\mathbb{P}}={\mathbb{P}}^{N}_{(x_{0}:\cdots:x_{N})}$ as the affine chart $(x_{0}\neq 0)$, the polynomials $f_{1},\ldots,f_{N}$ are represented by polynomials $F_{1},\ldots,F_{N}$, where $F_{i}(o)=0$. We have to estimate the codimension of the subset of tuples $(F_{*})$, for which there exists an irreducible subvariety $Y\ni o$ of positive dimension, which is not a line and such that $F_{i}_{Y}\equiv 0$ for all $i=1,\ldots,N$. For every such tuple there is a uniquely determined integer $k\in\{0,1,\ldots,N1\}$, satisfying the two conditions:

•
${\rm codim}_{o}(\{F_{1}=\cdots=F_{k}=0\}\subset{\mathbb{P}})=k$ (where ${\rm codim}_{o}$ means the codimension in a neighborhood of the point $o$, and for $k=0$ the set $\{F_{1}=\cdots=F_{k}=0\}$ is the whole space ${\mathbb{P}}$),

•
the polynomial $F_{k+1}$ vanishes identically on an irreducible component $B$ of the closed set $\{F_{1}=\cdots=F_{k}=0\}$, containing the point $o$, and if $k=N1$, then $B$ is not a line.
Let
$\displaystyle\alpha_{k}=\sum^{k+1}_{i=1}{d_{i}+Nk\choose d_{i}}k(Nk)$ 
be the codimension of the closed set of such tuples $(F_{1},\ldots,F_{k+1})$, that $F_{i}_{\Lambda}\equiv 0$ for a certain linear subspace $\Lambda\subset{\mathbb{P}}$ of codimension $k$, where $k=0,1,\ldots,N2$ (in order to see that the codimension of this closed set is indeed equal to $\alpha_{k}$, one argues as in the proof of Proposition 2.1: consider the algebraic set
$\displaystyle\{(\Lambda,\underline{F})\,\,\underline{F}_{\Lambda}\equiv 0\} \subset G(k,N)\times\{(\underline{F})\},$ 
where $G(k,N)$ is the projective Grassmanian of $k$subspaces in ${\mathbb{P}}^{N}$, and two projections on the direct factors $G(k,N)$ and the space $\{(\underline{F})\}=\{(F_{1},\ldots,F_{N})\}$ of tuples of homogeneous polynomials, introduced above; the obvious details are left to the reader). It is easy to check that $\alpha_{k}\geqslant d_{1}N+1$. Therefore, estimating the codimension of the set of “irregular” tuples $(F_{*})$, we may assume that the irreducible component $B$ of the closed set $\{F_{1}=\cdots=F_{k}=0\}$, on which $F_{k+1}$ vanishes identically, is not a linear subspace. Set
$\displaystyle\beta_{k}={\rm min}_{l\in\{1,\ldots,k\}}[(d_{1}+\cdots+d_{kl}+d_ {k+1}(kl))(Nk+l)+1],$ 
$k=1,\ldots,N1$. Now the technique developed in Pukhlikov ([Pukhlikov2001], Section 3) gives the estimate
$\displaystyle{\rm codim}((\overline{X_{\infty}\backslash X_{\rm line}})\subset {\cal P}(\underline{d}))\geqslant{\rm min}_{k\in\{1,\ldots,N1\}}\beta_{k},$ 
so that in order to complete the proof of Proposition 2.1 it is sufficient to show that the righthand side of the last inequality is not smaller than $(d_{1}N+1)$. This is an easy task.
Now replacing in the expression for $\beta_{k}$ the numbers $d_{1},\ldots,d_{k+1}$ by $d=d_{1}$, we get
$\displaystyle\beta_{k}\geqslant{\rm min}_{l\in\{1,\ldots,k\}}[((kl+1)d(kl)) (Nk+l)+1].$ 
The expression in the square brackets is a quadratic polynomial in $l$ with the senior coefficient $(d1)l^{2}$. Since $d\geqslant 2$, the minimum is attained at one of the endpoints of the interval $[1,k]$. For $l=k$ we get the value $dN+1$, which is what we want. For $l=1$ we get
$\displaystyle(k(d1)+1)(Nk+1)+1.$ 
Here $k\in\{1,\ldots,N1\}$ and the last expression is again a quadratic polynomial with the senior coefficient $(d1)k^{2}$, that is, the minimum in $k$ is attained at one of the endpoints of the interval $[1,N1]$. For $k=1$ we get the required value $dN+1$. For $k=N1$ we get $2d(N1)2N+5\geqslant dN+1$. This completes the proof of Proposition 2.2. $\square$
Corollary 2.1.
Assume that
$\displaystyle a\leqslant{\rm min}\left(d_{1}N,\sum^{N}_{i=1}d_{i}N\right).$ 
Then the number $\mu(a)$ is finite.
Below we obtain estimates from above for $\mu(a)$, which are close to optimal ones, for the values $a\leqslant N$.
2.3 The Rank of a System of Linear Forms
Let us consider the construction of the Example 1.6 more formally.
Example 2.1.
(See [Pukhlikov2004], Section 3.5) For $b\in{\mathbb{Z}}_{+}$ set
$\displaystyle X(b)=\{\underline{f}\in{\cal P}(\underline{d})\,\,{\rm rk}(df_{ 1}(o),\ldots,df_{N}(o))\leqslant Nb\}.$ 
For $b\leqslant N$ the set $X(b)$ is nonempty, closed and biinvariant, and of codimension
$\displaystyle{\rm codim}(X(b)\subset{\cal P}(\underline{d}))=b^{2}.$ 
For a general tuple $\underline{f}$ in any irreducible component of the set $X(b)$ there is a subset $I\subset\{1,\ldots,N\}$, $\sharp I=Nb$, such that
$\displaystyle{\rm rk}(df_{i}(o)\,\,i\in I)=Nb.$ 
Therefore, for any polynomials $g_{j}\in{\cal P}_{[1,d_{j}],N}$, $j\not\in I$, such that
$\displaystyle dg_{j}(o)\in\langle df_{i}(o)\,\,i\in I\rangle,$ 
the tuple $(f^{*}_{1},\ldots,f^{*}_{N})$, given by the conditions

•
$f^{*}_{i}=f_{i}$ for $i\in I$,

•
$f^{*}_{j}=g_{j}$ for $j\not\in I$,
belongs to the same irreducible component of the set $X(b)$, as $\underline{f}$. In other words, the closed algebraic set $Z_{I}(\underline{f})=\{f_{i}=0\,\,i\in I\}$ in a neighborhood of the point $o\in{\mathbb{A}}$ is a nonsingular $b$dimensional variety, and on the polynomials $f_{j},j\not\in I$, only one condition is imposed: $df_{j}_{T_{o}Z_{I}(\underline{f})}\equiv 0$. Therefore, for every irreducible component $B\subset X(b)$ we have
$\displaystyle\mu(B)=2^{b}.$ 
Since ${\rm codim}(B\subset{\cal P}(\underline{d}))\geqslant b^{2}$, we obtain the following estimate for the function $\mu(a)$ from below:
$\displaystyle\mu(a)\geqslant 2^{[\sqrt{a}]}.$ 
This example motivates introducing a new parameter that characterizes an arbitrary irreducible subvariety $B\subset{\cal P}(\underline{d})$ of codimension $a\in{\mathbb{Z}}_{+}$: set
$\displaystyle\beta(B)=N\max\limits_{\underline{f}\in B}{\rm rk}(df_{i}(o)\, \,i=1,\ldots,N).$ 
Obviously, $\beta(B)={\rm max}\{b\in{\mathbb{Z}}_{+}\,\,B\subset X(b)\}$. In particular, $\beta(B)\leqslant\sqrt{a}$.
Proposition 2.3.
If $\beta(B)=0$, then $\mu(B)=1$.
Proof.
This is obvious: for a general tuple $\underline{f}\in B$ the linear forms $df_{i}(o)$ are linearly independent. Q.E.D. for the proposition. $\square$
Proposition 2.4.
The following equality holds: $\mu(1)=2$.
Proof.
Let $B\subset{\cal P}(\underline{d})$ be an irreducible subvariety of codimension 1. If $\beta(B)=0$, then $\mu(B)=1$. Assume that $\beta(B)=1$. This means that for a general tuple $\underline{f}\in B$ there is an index $i$, such that the linear forms
$\displaystyle df_{1}(o),\ldots,df_{i1}(o),\quad df_{i+1}(o),\ldots,df_{N}(o)$ 
are linearly independent, so that the set
$\displaystyle\{f_{j}=0\,\,j=1,\ldots,i1,i+1,\ldots,N\}$ 
in a neighborhood of the point $o$ is a curve $C(\underline{f})$, which is nonsingular at the point $o$, and moreover $df_{i}_{T_{o}C(\underline{f})}\equiv 0$. Since ${\rm codim}(B\subset{\cal P}(\underline{d}))=1$, for any polynomial $g\in{\cal P}_{[1,d_{i}],N}$, such that $g_{T_{o}C(\underline{f})}\equiv 0$, we have $(f_{1},\ldots,f_{i1},g,f_{i+1},\cdots f_{N})\in B$. In particular, this is true for any quadratic form $g\in{\cal P}_{2,N}$. Therefore, for a general tuple $\underline{f}\in B$ we have $\mu(\underline{f})=2$, as we claimed. Q.E.D. for the proposition. $\square$
3 Reduction to a Smaller Dimension
In this section we construct an inductive procedure of estimating the multiplicity $\mu(B)$ in terms of multiplicities $\mu(B_{i})$ for certain subvarieties $B_{i}$ in the space of tuples of $(N1)$ polynomials in $(N1)$ variables. Iterating this procedure, we obtain in Sect. 4 estimates for the function $\mu(a)$. In Sect. 3.1 we construct the map of bringing a general tuple $(\underline{f})\in B$ into the standard form, in Sect. 3.2 we state the main claim about reduction to a smaller dimension, in Sects. 3.3–3.4 we prove it, in Sect. 3.5 we consider its generalization.
3.1 Bringing into the Standard Form
Let $B\subset{\cal P}(\underline{d})$ be an irreducible biinvariant subvariety of codimension $a$ and $\beta(B)=b\geqslant 1$; as we have seen, $a\geqslant b^{2}$. Consider the subset
$\displaystyle I\subset\{1,\ldots,N\},$ 
$\sharp I=b$, such that for a general tuples $\underline{f}\in B$
${\rm rk}(df_{i}(o)\,\,i\not\in I)=Nb.$  (3) 
Because of the biinvariance of the set $B$ we may assume that the following conditions are satisfied:

•
for $j\in I$ the linear form $df_{j}(o)$ is a linear combination of the forms $df_{i}(o)$, where $i\not\in I$ and $i<j$;

•
for $d_{j}=d_{j+1}$ if $j\in I$, then $(j+1)\in I$.
By the symbol $B^{\circ}$ we denote the open subset in $B$, defined by the condition (3). Set $e={\rm max}\{j\,\,j\in I\}$. Consider the space of tuples of polynomials
$\displaystyle\widetilde{\cal P}(\underline{d})=\prod_{j\not\in I}{\cal P}_{[1, d_{j}],N}\times\prod_{j\in I}{\cal P}_{[2,d_{j}],N}.$ 
On the open set $B^{\circ}$ the map of bringing into the standard form $\sigma\colon B^{\circ}\to\widetilde{\cal P}(\underline{d})$,
$\displaystyle\sigma\colon(f_{1},\ldots,f_{N})\mapsto(\tilde{f}_{1},\cdots \tilde{f}_{N}),$ 
is well defined, where $\tilde{f}_{j}=f_{j}$ for $j\not\in I$ and $\tilde{f}_{j}=f_{j}\sum_{i\not\in I,i<j}\lambda_{j,i}f_{i}$, the coefficients $\lambda_{i}\in{\mathbb{C}}$ are uniquely determined by the relation
$\displaystyle df_{j}(o)=\sum_{i\not\in I,i<j}\lambda_{j,i}df_{i}(o).$ 
Set $B^{\rm st}=\overline{\sigma(B^{\circ})}\subset\widetilde{\cal P}(\underline{d})$. Setting for $j=1,\ldots,N$
$\displaystyle\varepsilon(j)=\sharp\{i\not\in I,i<j\},$ 
we see that the fibre of general position $\sigma\colon B^{\circ}\to B^{\rm st}$ is of dimension
$\displaystyle\sum_{j\in I}\varepsilon(j)\leqslant b(Nb)$ 
(this inequality becomes an equality if $I=\{Nb+1,\ldots,N\}$). Taking into account that
$\displaystyle{\rm dim}\widetilde{\cal P}(\underline{d})={\rm dim}{\cal P}( \underline{d})bN,$ 
we conclude that
$\displaystyle{\rm codim}(B^{\rm st}\subset\widetilde{\cal P}(\underline{d}))=a bN+\sum_{j\in I}\varepsilon(j)\leqslant ab^{2}.$ 
Now let us represent the space $\widetilde{\cal P}(\underline{d})$ as the direct product of the spaces
$\displaystyle\widetilde{\cal P}_{e}(\underline{d})=\prod_{j\not\in I}{\cal P}_ {[1,d_{j}],N}\times\prod_{j\in I,j\neq e}{\cal P}_{[2,d_{j}],N}$ 
and ${\cal P}_{[2,d_{e}],N}$. The projections onto these direct factors denote by the symbols $\pi$ and $\pi_{e}$, respectively.
Let us consider the restriction $\pi_{B}=\pi_{B^{\rm st}}\colon B^{\rm st}\to\widetilde{{\cal P}}_{e}( \underline{d})$ of the projection $\pi$ onto $B^{\rm st}$. For a general tuple $\underline{f}\in B^{\rm st}$ set
$\displaystyle\gamma={\rm codim}\left(\pi^{1}_{B}(\pi(\underline{f}))\subset{ \cal P}_{[2,d_{e}],N}\right).$ 
(Here we identify the fibre $\pi^{1}(\pi(\underline{f}))$ and the space ${\cal P}_{[2,d_{e}],N}$.) Therefore, the equality
$\displaystyle{\rm codim}(\overline{\pi(B^{\rm st}})\subset\widetilde{\cal P}_{ e}(\underline{d}))=abN+\sum_{j\in I}\varepsilon(j)\gamma$ 
holds, whereas the right hand side does not exceed $ab^{2}\gamma$, so that the estimate
$b^{2}+\gamma\leqslant a$  (4) 
holds. Before studying the just constructed map $\pi_{B}$ in full generality, let us consider some simple examples.
Example 3.1.
Assume that $b=1$ and $\gamma=0$, that is, every nonempty fibre of the map $\pi_{B}\colon B^{\rm st}\to\widetilde{\cal P}_{e}(\underline{d})$ is the whole space of polynomials ${\cal P}_{[2,d_{e}],N}$. Since $b=1$, for a general tuple $\underline{f}\in B^{\circ}$ the closed set $C=\{f_{i}=0\,\,i\neq e\}$ in a neighborhood of the point $o$ is a nonsingular curve and the polynomial $f_{e}$ is of the form
$g(z_{*})+\sum_{i\neq e}\lambda_{i}f_{i},$  (5) 
where $g\in{\cal P}_{[2,d_{e}],N}$ is an arbitrary polynomial (as bringing into the standard form means subtracting from $f_{e}$ a linear combination of polynomials $f_{i}$, $i\neq e$, and gives $g(z_{*})$ as a result).
Therefore, the restriction $f_{e}_{C}$ has at the point $o$ a zero of order exactly 2, that is, the equality $\mu(B)=2$ holds. This equality does not depend on the codimension $a$ of the subvariety $B$.
Example 3.2.
Now let us assume that $b=\gamma=1$, and $d_{e}\geqslant 3$. This case is not much more complicated. Again for a general tuple $\underline{f}\in B^{\circ}$ the set $C=\{f_{i}=0\,\,i\neq e\}$ is a curve, nonsingular at the point $o$, and the polynomial $f_{e}$ is of the form (5), where in this case
$\displaystyle g\in\pi^{1}_{B}(\pi(\underline{f}))\subset{\cal P}_{[2,d_{e}],N}$ 
(again, identifying the fibre of the projection $\pi$ with the space ${\cal P}_{[2,d_{e}],N}$ by means of the projection $\pi_{e}$, which is meant but not written) and $\pi^{1}_{B}(\pi(\underline{f}))$ is of codimension 1 in the ambient space ${\cal P}_{[2,d_{e}],N}$. Blowing up the point $o\in{\mathbb{A}}$, it is easy to see that for $d_{e}\geqslant 3$ the condition
$\displaystyle{\rm ord}_{o}g_{C}\geqslant 3$ 
defines an irreducible divisor in ${\cal P}_{[2,d_{e}],N}$, and moreover for a general polynomial $g$ in that divisor the equality ${\rm ord}_{o}g_{C}=3$ holds. Therefore, in the case under consideration we have $\mu(B)\in\{2,3\}$ for any value $a$ of the codimension of the subvariety $B$.
3.2 Splitting Off a Direct Factor
Now let us consider the general case. Let $\underline{d}^{+}=(d_{1},\ldots,d_{e1},d_{e+1},\ldots,d_{N})$ be the truncated tuple of degrees and
$\displaystyle{\cal P}(\underline{d}^{+})=\prod_{j\neq e}{\cal P}_{[1,d_{j}],N1}$ 
the corresponding space of tuples of $(N1)$ polynomials in $(N1)$ variables. We keep the notations introduced at the beginning os Sect. 3.1: $B\subset{\cal P}(\underline{d})$ is an irreducible subvariety of codimension $a$ with $\beta(B)=b$.
Theorem 3.1.
Assume that $\gamma\leqslant N1$. Then there are irreducible biinvariant subvarieties $B_{i}\subset{\cal P}(\underline{d}^{+})$, $i=1,2$, such that:

(i)
the inequality $\mu(B)\leqslant\mu(B_{1})+\mu(B_{2})$ holds,

(ii)
$\beta(B_{1})=b1$ and the estimate
$a_{1}={\rm codim}(B_{1}\subset{\cal P}(\underline{d}^{+}))\leqslant a2b+1$ (6) holds,

(iii)
either $\beta(B_{2})=b1$ and the codimension $a_{2}={\rm codim}(B_{2}\subset{\cal P}(\underline{d}^{+}))$ satisfies the inequality (6) (in this case we say that this is the case of stable reduction), or $\beta(B_{2})=b$ and the inequality
$a_{2}\leqslant ab$ (7) holds (in that case we say that this is the case of nonstable reduction).
Before starting to show the theorem, let us explain briefly the strategy of the proof. Bringing the tuples of the subvariety $B$ into the standard form, we obtained a new subvariety $B^{\rm st}$, where the $e$th polynomial of every tuple has no linear term. Now we intersect $B^{\rm st}$ with a special subvariety of tuples, the $e$th polynomial in which is a reducible quadratic form (a product of two linear forms), whereas the other polynomials are arbitrary. Calculating dimensions and taking into account the biinvariance of $B$, we show that the intersection is nonempty and estimate its (co)dimension. Now we can use the following obvious observation: the multiplicity at zero of a tuple $(f_{1},\ldots,f_{e},\ldots,f_{N})$ with $f_{e}$ just a product of two linear forms, say $h_{1}(z)h_{2}(z)$, is equal to the sum of multiplicities at zero of the tuples
$\displaystyle(f_{1}_{\{h_{i}=0\}},\ldots,f_{e1}_{\{h_{i}=0\}},f_{e+1}_{\{h _{i}=0\}},\ldots,f_{N}_{\{h_{i}=0\}}),$ 
$i=1,2$. The subvarieties $\{h_{i}=0\}$ are hyperplanes and can be identified with ${\mathbb{C}}^{N1}$. Therefore, we can estimate the original multiplicity $\mu(B)$ in terms of the multiplicities $\mu(B_{i})$, $i=1,2$, where the subvarieties $B_{i}$ consist of $(N1)$uples of polynomials in $(N1)$ variables. (It turns out that one of the linear forms, say $h_{1}(z)$, can be preselected.) The main work in the proof is to estimate the parameters of the new subvarieties $B_{i}$.
Now we proceed to the rigorous argument.
Proof of Theorem 3.1.
Let $h_{1}(z_{*})\in{\cal P}_{1,N}$ be a linear form of general position with respect to the subset $\pi(B^{\rm st})\subset\widetilde{{\cal P}_{e}}(\underline{d})$, in the sense that the hyperplane $\{h_{1}=0\}$ does not contain the linear space
$\displaystyle\{df_{j}(o)=0\,\,j\not\in I\}$ 
for a general tuple $\underline{f}\in B$. Furthermore, let
$\displaystyle\Pi=\{h_{1}(z_{*})h(z_{*})\,\,h\in{\cal P}_{1,N}\}\subset{\cal P }_{2,N}$ 
be the linear space of reducible homogeneous quadratic polynomials divisible by $h_{1}$. Note that ${\cal P}_{2,N}\subset{\cal P}_{[2,d_{e}],N}$, so that we may (and will) consider $\Pi$ as a linear subspace in ${\cal P}_{[2,d_{e}],N}$. Obviously, ${\rm dim}\Pi=N$. Set
$\displaystyle{\cal P}_{\Pi}=\pi^{1}_{e}(\Pi)=\widetilde{{\cal P}}(\underline{ d})\times\Pi.$ 
This is closed irreducible subset of the space $\widetilde{{\cal P}}(\underline{d})$. The following claim is true.
Proposition 3.1.
The intersection $B^{\rm st}\cap{\cal P}_{\Pi}$ is nonempty and is of codimension at most $abN+\sum_{j\in I}\varepsilon(j)\leqslant ab^{2}$ in ${\cal P}_{\Pi}$. Moreover, the equality
$\displaystyle\overline{\pi(B^{\rm st})}=\overline{\pi(B^{\rm st}\cap{\cal P}_{ \Pi})}$ 
holds, and for a general tuple $\underline{f}\in B^{\rm st}$ the intersection $\pi^{1}_{B}(\pi(\underline{f}))\cap{\cal P}_{\Pi}$ has positive dimension.
Proof.
As the set $B$ is biinvariant, for a general tuple $\underline{f}\in B^{\rm st}$ the fibre $\pi^{1}_{B}(\pi(\underline{f}))$ is a closed subset of the space ${\cal P}_{[2,d_{e}],N}$ of codimension $\gamma\leqslant N1$, containing the zero polynomial. Therefore, the intersection $\pi^{1}_{B}(\pi(\underline{f}))\cap{\cal P}_{\Pi}$ is a nonempty closed subset in $\Pi$ of codimension at most $\gamma$, so that its dimension is positive. The other claims are now obvious. The proof is complete.
By the symbol $\pi_{\Pi}$ we denote the projection ${\cal P}_{\Pi}\to\Pi$, the fibre of which is the space $\widetilde{{\cal P}}_{e}(\underline{d})$. By the biinvariance (more precisely, the invariance with respect to the group $G_{1}$ of linear changes of variables) either
$\displaystyle\overline{\pi_{\Pi}(B^{\rm st}\cap{\cal P}_{\Pi})}=\Pi,$ 
or $\pi_{\Pi}(B^{\rm st}\cap{\cal P}_{\Pi})$ is the line $\{\lambda h^{2}_{1}(z_{*})\,\,\lambda\in{\mathbb{C}}$} in $\Pi$, and then $\gamma=N1$ (as $\Pi$ is $N$dimensional). The second case is simpler, let us start with the first one.
Let $h_{1}h_{2}\in\pi_{\Pi}(B^{\rm st}\cap{\cal P}_{\Pi})$ be a general polynomial, $h_{2}\not\in\{\lambda h_{1}\,\,\lambda\in{\mathbb{C}}\}$. The intersection
$\displaystyle B^{\rm st}\cap\pi^{1}_{\Pi}(h_{1}h_{2})\subset\widetilde{\cal P }_{e}(\underline{d})$ 
is a closed subset of codimension at most
$\displaystyle abN+\sum_{j\in I}\varepsilon(j)\gamma.$ 
Let us fix isomorphisms of the hyperplanes $\{h_{i}=0\}$ and ${\mathbb{C}}^{N1}$ and let
$\displaystyle\rho_{i}\colon{\cal P}_{[k,l],N}\to{\cal P}_{[k,l],N1}$ 
be the restriction map $\rho_{i}(f)=f_{\{h_{i}=0\}}$, $i=1,2$. The same symbols $\rho_{i}$ will be used for the corresponding maps of the spaces of tuples of polynomials:
$\displaystyle\rho_{i}(f_{1},\ldots,f_{k})=(\rho_{i}(f_{1}),\ldots,\rho_{i}(f_{ k})).$ 
Omitting the polynomial $f_{e}$, we obtain two projections
$\displaystyle\rho_{i}\colon\widetilde{\cal P}_{e}(\underline{d})\to{\cal P}( \underline{d}^{+}),$ 
corresponding to restrictions onto the hyperplanes $H_{i}=\{h_{i}=0\}$. Finally, set
$\displaystyle B_{i}=\overline{\langle\rho_{i}\circ\pi(B^{\rm st}\cap\pi^{1}_{ \Pi}(h_{1}h_{2}))\rangle}\subset{\cal P}(\underline{d}^{+}),$ 
where the brackets $\langle\cdot\rangle$ mean the biinvariant span, and the line above means the closure. The sets $B_{i}$ without loss of generality can be assumed to be irreducible (if this is not the case, take any irreducible component).
Let us show the claim (i) of Theorem 3.1. For a general tuple of polynomials
$\displaystyle\underline{f}=(f_{1},\ldots,f_{e1},h_{1}h_{2},f_{e+1},\ldots,f_{ N})\in B^{\rm st}\cap\pi^{1}_{\Pi}(h_{1}h_{2})$ 
the inequality $\mu(\underline{f})\geqslant\mu(B)$ holds, because $\underline{f}\in B$. Let
$\displaystyle\underline{f}^{+}=(f_{1},\ldots,f_{e1},f_{e+1},\ldots,f_{N})$ 
be a truncated tuple. Obviously,
$\displaystyle\mu(\underline{f})=\mu(\rho_{1}(\underline{f}^{+}))+\mu(\rho_{2}( \underline{f}^{+})),$ 
where $\rho_{i}(\underline{f}^{+})$ are tuples of general position in the algebraic sets
$\displaystyle\rho_{i}\circ\pi(B^{\rm st}\cap\pi^{1}_{\Pi}(h_{1}h_{2})),$ 
so that $\mu(\rho_{i}(\underline{f}^{+}))=\mu(B_{i})$. This proves the claim (i). $\square$
3.3 Restriction onto the Hyperplane $H_{1}$
Let us show the claim (ii). By the generality of the linear form $h_{1}$, the linear forms $df_{i}(o)$, $i\not\in I$, remain linearly independent after being restricted on the hyperplane $H_{1}=\{h_{1}=0\}$:
$\displaystyle{\rm rk}(df_{i}(o)_{H_{1}}\,\,i\not\in I)=Nb,$ 
so that $\beta(B_{1})=b1$. Since the projection $\rho_{1}$ is a surjective linear map, the codimension of the set $\rho_{i}\circ\pi(B^{\rm st}\cap\pi^{1}_{\Pi}(h_{1}h_{2}))$ in the space
$\displaystyle\prod_{j\not\in I}{\cal P}_{[1,d_{j}],N1}\times\prod_{j\in I,j \neq e}{\cal P}_{[2,d_{j}],N1}$ 
does not exceed the codimension of the set $\pi(B^{\rm st}\cap\pi^{1}_{\Pi}(h_{1}h_{2}))$ in the space $\widetilde{\cal P}_{e}(\underline{d})$. Now let us apply the procedure, inverse to the procedure of bringing into the standard form: the variety $B_{1}$ with every tuple
$\displaystyle\underline{f}^{+}=(f_{1},\ldots,f_{e1},f_{e+1},\ldots,f_{N})\in \rho_{i}\circ\pi(B^{\rm st}\cap\pi^{1}_{\Pi}(h_{1}h_{2}))$ 
contains also all tuples $(\widetilde{f}_{1},\ldots,\widetilde{f}_{e1},\widetilde{f}_{e+1},\ldots, \widetilde{f}_{N})$, where $\widetilde{f}_{j}=f_{j}$ for $j\not\in I$ and
$\displaystyle\widetilde{f}_{j}=f_{j}+\sum_{i\not\in I,i<j}\lambda_{j,i}f_{i}$ 
for $j\in I$, $j\neq e$, for all possible tuples of coefficients $(\lambda_{*,*})$, and different tuples of coefficients determine different tuples of polynomials $(\underline{\widetilde{f}})$. Therefore, the codimension ${\rm codim}(B_{1}\subset{\cal P}(\underline{d}^{+}))$ is bounded from above by the number
$\displaystyle abN+\sum_{j\in I}\varepsilon(j)\gamma+(b1)(N1)\sum_{j\in I, j\neq e}\varepsilon(j)$  
$\displaystyle\quad=aNb\gamma+\varepsilon(e)+1.$ 
Taking into account that $\varepsilon(e)\leqslant\sharp\{j\not\in I\}=Nb$, we obtain the estimate (6). This completes the proof of the claim (ii).
Remark 3.1.
If $\gamma=0$, then $\pi^{1}_{B}(\pi(\underline{f}))$ is the whole fibre of the projection $\pi$, that is, the linear space ${\cal P}_{[2,d_{e}],N}$. In that case $h_{2}\in{\cal P}_{1,N}$ is any form of general position (in fact, in this case we could take $h_{2}=h_{1}$ and make no assumption that $h_{2}\neq\lambda h_{1}$). Therefore, $\mu(B)\leqslant 2\mu(B_{1})$, where $B_{1}\subset{\cal P}(\underline{d}^{+})$ is a subvariety with $\beta(B_{1})=b1$, the codimension of which satisfies the estimate (6). The claim (iii) in this case is not needed (obviously, this is the case of stable reduction).
3.4 Restriction onto the Hyperplane $H_{2}$
Let us show the claim (iii). For a general tuple of polynomials
$\displaystyle(f_{1},\ldots,f_{e1},f_{e+1},\ldots,f_{N})\in\pi(B^{\rm st}\cap \pi^{1}_{\Pi}(h_{1}h_{2}))$ 
there are two options:

•
the subspace $\{df_{i}(o)=0\,\,i\not\in I\}$ is not contained in the hyperplane $H_{2}=\{h_{2}=0\}$,

•
the subspace $\{df_{i}(o)=0\,\,i\not\in I\}$ is contained in $H_{2}$.
In the first case we have the stable reduction: $\beta(B_{2})=b1$ and, arguing in word for word the same way as in Sect. 3.3, we get that the codimension of the subvariety $B_{2}$ satisfies the inequality (6).
Therefore we assume that the second case takes place, so that
$\displaystyle{\rm rk}(df_{i}(o)_{H_{2}},i\not\in I)=Nb1$ 
and $\beta(B_{2})=b$. Between the linear forms $df_{i}(o)_{H_{2}},i\not\in I$, there is exactly one linear dependence, so that there is a unique index $m\not\in I$, satisfying the relation
$\displaystyle df_{m}(o)_{H_{2}}=\sum_{i<m,i\not\in I}\lambda_{i}df_{i}(o)_{H _{2}}$ 
with uniquely determined coefficients $\lambda_{i}$. Therefore, on the Zariski open subset
$\displaystyle(\rho_{2}\circ\pi(B^{\rm st}\cap\pi_{\Pi}(h_{1}h_{2})))^{\circ}$ 
(see the condition (3) at the beginning of Sect. 3.1) we have a well defined map $\sigma_{m}$ of bringing into the standard form in the $m$th factor:
$\displaystyle\sigma_{m}\colon(g_{i}\,\,i\neq e)\mapsto(\widetilde{g}_{i}\,\, i\neq e).$ 
The natural ambient space for the righthand side is
$\displaystyle\prod_{j\not\in I,j\neq m}{\cal P}_{[1,d_{j}],N1}\times{\cal P}_ {[2,d_{m}],N1}\times\prod_{j\in I,j\neq e}{\cal P}_{[2,d_{j}],N1}$ 
(up to a permutation of the direct factors), and the codimension of the closed subset
$\displaystyle\overline{\sigma_{m}\circ(\rho_{2}\circ\pi(B^{\rm st}\cap\pi^{1} _{\Pi}(h_{1}h_{2})))^{\circ}}$ 
with respect to that ambient space does not exceed
$\displaystyle abN+\sum_{j\in I}\varepsilon(j)\gamma(N1)+\varepsilon(m).$ 
Now we argue in a word for word the same way as when restricting onto the hyperplane $H_{1}$: we apply the procedure, inverse to the procedure of bringing into the standard form in the factors with numbers
$\displaystyle j\in(I\backslash\{e\})\cup\{m\}.$ 
The variety $B_{2}$ with every tuple
$\displaystyle\underline{g}^{+}=(g_{i}\,\,i\neq e)\in\overline{\sigma_{m}\circ (\rho_{2}\circ\pi(B^{\rm st}\cap\pi^{1}_{\Pi}(h_{1}h_{2})))^{\circ}}$ 
contains all tuples $(\widetilde{g}_{i}\,\,i\neq e)$, where $\widetilde{g}_{j}=g_{j}$ for $j\not\in I$, $j\neq m$ and
$\displaystyle\widetilde{g}_{j}=g_{j}+\sum_{i\not\in I,i\neq m,i<j}\lambda_{j,i }g_{i}$ 
for $j\in(I\backslash\{e\})\cup\{m\}$. Therefore, the codimension ${\rm codim}(B_{2}\subset{\cal P}(\underline{d}^{+}))$ is bounded from above by the number
$\displaystyle abN+\sum_{j\in I}\varepsilon(j)\gamma(N1)+\varepsilon(m)+b(N 1)$  
$\displaystyle\qquad\sum_{j\in I,j\neq e,j<m}\varepsilon(j)\sum_{j\in I,j\neq e ,j>m}(\varepsilon(j)1)\varepsilon(m)$  
$\displaystyle\quad=ab\gamma(N1)+\varepsilon(e)+\sharp\{j\in I\backslash\{e \}\,\,j>m\}.$ 
Since obviously $\sharp\{j\in I\backslash\{e\}\,\,j>m\}\leqslant b1$ and $\varepsilon(e)\leqslant Nb$, this implies the inequality (7). Proof of the claim (iii) is complete.
In the beginning of the proof of Theorem 3.1 we put off the case when $\pi_{\Pi}(B^{\rm st}\cap{\cal P}_{\Pi})$ is the line $\langle h^{2}_{1}(z_{*})\rangle$, so that $\gamma=N1$. In that case $H_{1}=H_{2}$, so that we have stable reduction (see Remark 3.1).
Proof of Theorem 3.1 is complete. Q.E.D.$\square$
3.5 The Case of High Codimension
Now let us assume that $\gamma\geqslant N$. In that case it is easy to state and prove an analog of Theorem 3.1; however, with $\gamma$ growing the resulting estimates get less and less useful.
Set $k=\left[\frac{\gamma}{N}\right]+1\geqslant 2$.
Proposition 3.2.
Assume that $d_{e}\geqslant k$. Then there are irreducible biinvariant subvarieties $B_{i}\subset{\cal P}(\underline{d}^{+})$, $i=1,\ldots,k$, such that

(i)
the inequality $\mu(B)\leqslant\mu(B_{1})+\cdots+\mu(B_{k})$ holds ,

(ii)
for every $i\in\{1,\ldots,k\}$ we have: either $\beta(B_{i})=b1$, and then the codimension $a_{i}={\rm codim}(B_{i}\subset{\cal P}(\underline{d}^{+}))$ satisfies the inequality (6), or $\beta(B_{i})=b$ and the codimension $a_{i}$ satisfies the inequality (7).
Proof.
repeats the proof of Theorem 3.1 word for word, with only one difference: for $\Pi$ we have to take the irreducible subvariety of decomposable forms of degree $k$,
$\displaystyle\Pi=\left\{\prod^{k}_{j=1}h_{j}(z_{*})\,\,h_{j}\in{\cal P}_{1,N} \right\}\subset{\cal P}_{k,N}\subset{\cal P}_{[2,d_{e}],N}$ 
(the last inclusion is provided by our assumption that $d_{e}\geqslant k$). Obviously, ${\rm dim}\Pi>\gamma$, so that the arguments used in the proof of Theorem 3.1 work in this case. Setting
$\displaystyle B_{j}=\overline{\langle\rho_{j}\circ\pi(B^{\rm st}\cap\pi^{1}_{ \Pi}(h_{1}\cdots h_{k}))\rangle}\subset{\cal P}(\underline{d}^{+}),$ 
where $\rho_{j}$ is the restriction onto the hyperplane $H_{j}=\{h_{j}=0\}$, and all the symbols that we use have the same meaning as in Sects. 3.2–3.4, we obtain the inequality
$\displaystyle\mu(B)\leqslant\mu(B_{1})+\cdots+\mu(B_{k}).$ 
Repeating the arguments of Sects. 3.3, 3.4, we obtain the claim (ii). This completes the proof of Proposition 3.2.
The proposition that we have just shown is far from being as useful as Theorem 3.1, because $k$ can be high and, the main point, all subvarieties $B_{i}$ can have $\beta(B_{i})=b$, which essentially weakens the estimates obtained by iterating Proposition 3.2. $\square$
4 Explicit Estimates for Multiplicities
In this section we obtain estimates for $\mu(a)$ for $a\leqslant N$, which are close to optimal ones. First, we consider the case of a subvariety $B\subset{\cal P}(\underline{d})$ with $\beta(B)=1$ as an example, when it is easy to obtain a precise estimate from above for $\mu(B)$. Then using Theorem 3.1, we construct a recurrent procedure of estimating the multiplicity, based on controlling two parameters, the codimension $a$ and $b=\beta(B)$. (Recall that $a\geqslant b^{2}$.) At first this procedure is applied to obtain the estimates for small values of the codimension $a\leqslant 49$. After that, we consider the general case: in Sects. 4.3–4.5 we prove estimates from above for $\mu(a)$, where the estimating function grows as $C^{\sqrt{a}}$, here $C>0$ is some effectively estimated constant.
4.1 Estimating the Multiplicity for $b=1$
Let ${\cal P}(\underline{d})$ be an irreducible biinvariant subvariety of codimension $a\leqslant N$.
Proposition 4.1.
Assume that $\beta(B)=1$. Then the inequality $\mu(B)\leqslant a+1$ holds.
Proof.
As we saw above (Proposition 2.4), for $a=1$ we have $\mu(B)\leqslant 2$. Therefore we may assume that $N\geqslant a\geqslant 2$ and prove the proposition by induction on $N$. In the notations of Sect. 3, we have $\gamma\leqslant a1\leqslant N1$, so that we can apply Theorem 3.1: there are irreducible biinvariant subvarieties $B_{i}\subset{\cal P}(\underline{d}^{+})$, such that $\mu(B)\leqslant\mu(B_{1})+\mu(B_{2})$. Here $\beta(B_{1})=0$, so that $\mu(B_{1})=1$. On the other hand, $a_{2}={\rm codim}(B_{2}\subset{\cal P}(\underline{d}^{+}))\leqslant a1$, so that by the inductive assumption
$\displaystyle\mu(B_{2})\leqslant a_{2}+1\leqslant a.$ 
Q.E.D. for the proposition. $\square$
Remark 4.1.
The estimate in Proposition 4.1 is sharp: for any $a\leqslant N$ there is an irreducible biinvariant subvariety $B\subset{\cal P}(\underline{d})$ of codimension $a$ with $\beta(B)=1$ and $\mu(B)=a+1$. Indeed, let $B^{\circ}\subset{\cal P}(\underline{d})$ be defined by the conditions

•
the equality ${\rm rk}(df_{1}(o),\ldots,df_{N1}(o))=N1$ holds, so that the set $C=\{f_{1}=\cdots=f_{N1}=0\}$ in a neighborhood of the point $o$ is a curve, nonsingular at that point,

•
the inequality ${\rm ord}_{o}\left(f_{N}_{C}\right)\geqslant a+1$ holds.
It is easy to see that ${\rm codim}(B^{\circ}\subset{\cal P}(\underline{d}))\leqslant a$, so that for the closure $B$ of the biinvariant span $\langle B^{\circ}\rangle$ the more so ${\rm codim}(B\subset{\cal P}(\underline{d}))\leqslant a$, and $\mu(B)\geqslant a+1$. Therefore, the last two inequalities we have the equality (the strict inequalities are impossible by Proposition 4.1).
Let us show now that if the degrees $d_{i}$ are high enough, then for $\beta(B)=1$ the restriction $a\leqslant N$ for the codimension is not needed.
Proposition 4.2.
Assume that $d_{1}\geqslant a+1$ and $\beta(B)=1$. Then $\mu(B)\leqslant a+1$.
Proof.
Let $\underline{f}\in B$ be a general tuple. For some $e\in\{1,\ldots,N\}$ we have:
$\displaystyle{\rm rk}(df_{i}(o)\,\,i\neq e)=N1,$ 
so that $C=\{f_{i}=0\,\,i\neq e\}$ in a neighborhood of the point $o$ is a curve, nonsingular at that point. It is sufficient to show that the condition ${\rm ord}_{o}\left(f_{e}_{C}\right)\geqslant a+1$ defines a closed subset in ${\cal P}_{[1,d_{e}],N}$ of codimension at least $a$ (and for that reason precisely $a$). But this is obvious: let $l(z_{*})$ be a general linear form, then the polynomials $l^{i}(z_{*})$, $i=1,\ldots,a+1$, satisfy the condition ${\rm ord}_{o}\left(l^{i}(z_{*})_{C}\right)=i$. Therefore, the linear subspaces
$\displaystyle\Delta_{i}=\{f_{e}\in{\cal P}_{[1,d_{e}],N}\,\,{\rm ord}_{o} \left(f_{e}_{C}\right)\geqslant i\}$ 
are distinct and $\Delta_{1}={\cal P}_{[1,d_{e}],N}\supset\Delta_{2}\supset\cdots\supset\Delta_{ a+1}$ (recall that $d_{e}\geqslant d_{1}\geqslant a+1$). Therefore indeed ${\rm codim}(\Delta_{a+1}\subset\Delta_{1})\geqslant a$, as we need. Q.E.D. for the proposition. $\square$
Remark 4.2.
It seems that the assumption $d_{1}\geqslant a+1$ can be considerably relaxed.
4.2 Estimating Multiplicities for Small Codimensions
Theorem 3.1 shows that in order to estimate the multiplicity $\mu(B)$ one needs to take into account the value of the parameter $\beta(B)=b$. Let $U\subset{\mathbb{Z}}_{+}\times{\mathbb{Z}}_{+}$ be the set $\{(a,b)\,\,a\geqslant b^{2}\}$. Let us define inductively the function
$\displaystyle\overline{\mu}\colon U\to{\mathbb{Z}}_{+},$ 
setting $\overline{\mu}(a,0)\equiv 1$, $\overline{\mu}(a,1)\equiv a+1$, for $a<b(b+1)$
$\displaystyle\overline{\mu}(a,b)=2\overline{\mu}(a(2b1),b1),$ 
for $a\geqslant b(b+1)$
$\displaystyle\overline{\mu}(a,b)=\overline{\mu}(a(2b1),b1)+{\rm max}\{ \overline{\mu}(a(2b1),b1),\overline{\mu}(ab,b)\}.$ 
Theorem 3.1 immediately implies
Theorem 4.1.
Let $B\subset{\cal P}(\underline{d})$ be an irreducible biinvariant subvariety of codimension $a\leqslant N$ and $\beta(B)=b$. Then the inequality
$\displaystyle\mu(B)\leqslant{\bar{\mu}}(a,b)$ 
holds. In particular, $\mu(a)\leqslant\max_{0\leqslant b\leqslant\sqrt{a}}{\bar{\mu}}(a,b)$.
For small values of $a$ the function $\overline{\mu}$ is easy to compute by hand; it is not hard to write a computer program, computing $\overline{\mu}$, either. Below we give the table of values $\overline{\mu}(a,b)$ for $a\leqslant 49$, $b\leqslant 7$. The symbol $*$ means that the pair $(a,b)\not\in U$ and the value of the function $\overline{\mu}$ is not defined. Already for these small values of the codimension the growth of the values $\overline{\mu}(a,b)$ can be clearly seen. In boldface we give the maximal value $\overline{\mu}(a,b)$ for a given $a$.
$a$  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  

$b=0$  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  
$b=1$  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  
$b=2$  *  *  *  4  6  8  11  14  18  22  27  32  38  44  51  58  
$b=3$  *  *  *  *  *  *  *  *  8  12  16  22  28  36  44  55  
$b=4$  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  16  
$b=5$  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  * 
$a$  17  18  19  20  21  22  23  24  25  26  27  28  

$b=0$  1  1  1  1  1  1  1  1  1  1  1  1  
$b=1$  18  19  20  21  22  23  24  25  26  27  28  29  
$b=2$  66  74  83  92  102  112  123  134  146  158  171  184  
$b=3$  68  82  99  119  140  165  193  223  257  295  335  380  
$b=4$  24  32  44  56  72  88  110  136  164  198  238  280  
$b=5$  *  *  *  *  *  *  *  *  32  48  64  88  
$b=6$  *  *  *  *  *  *  *  *  *  *  *  * 
$a$  29  30  31  32  33  34  35  36  37  38  39  

$b=0$  1  1  1  1  1  1  1  1  1  1  1  
$b=1$  30  31  32  33  34  35  36  37  38  39  40  
$b=2$  198  212  227  242  258  274  291  308  326  344  363  
$b=3$  429  481  538  600  665  736  812  892  978  1070  1166  
$b=4$  330  391  461  537  625  726  841  966  1106  1264  1441  
$b=5$  112  144  176  220  272  328  396  476  560  660  782  
$b=6$  *  *  *  *  *  *  *  64  96  128  176  
$b=7$  *  *  *  *  *  *  *  *  *  *  * 
$a$  40  41  42  43  44  45  46  47  48  49  

$b=0$  1  1  1  1  1  1  1  1  1  1  
$b=1$  41  42  43  44  45  46  47  48  49  50  
$b=2$  382  402  422  443  464  486  508  531  554  578  
$b=3$  1269  1378  1492  1613  1741  1874  2015  2163  2317  2479  
$b=4$  1631  1842  2076  2333  2609  2912  3242  3602  3987  4404  
$b=5$  922  1074  1250  1452  1682  1932  2212  2528  2893  3313  
$b=6$  224  288  352  440  544  656  792  952  1120  1320  
$b=7$  *  *  *  *  *  *  *  *  *  128  
$b=8$  *  *  *  *  *  *  *  *  *  * 
Now let us consider the problem of obtaining a simple effective upper bound for the multiplicities $\mu(B)$. From the technical viewpoint, one needs to find a simple and visual formalization of the procedure of estimating these numbers in terms of the numbers $\mu(B^{\prime})$ for subvarieties $B^{\prime}\subset{\cal P}(\underline{d}^{\prime})$ in the spaces of truncated tuples $(d^{\prime}_{1},\ldots,d^{\prime}_{N^{\prime}})$ with $N^{\prime}<N$.
4.3 The General Method of Estimating the Multiplicity
The symbol $B$ stands for an irreducible biinvariant subvariety of codimension $a$ with $\beta(B)=b$. Let us consider the threeletter alphabet $\{A,C_{0},C_{1}\}$. Let ${\cal W}$ be the set of all words in that alphabet, including the empty word $\emptyset$. The length of the word $w$ is denoted by the symbol $w\in{\mathbb{Z}}_{+}$. The length of the empty word is equal to zero.
Let us describe a procedure of constructing a sequence of subsets $W_{l}\subset{\cal W}$, $l=0,1,\cdots$. The length of every word
$\displaystyle w\in\mathop{\bigcup}\limits_{l\in{\mathbb{Z}}_{+}}W_{l}$ 
does not exceed $N$. Set $N(w)=Nw\in{\mathbb{Z}}_{+}$. This sequences stabilizes, that is, $W_{l}=W_{l+1}$, starting from some $l=L$. For every word $w\in\cup W_{l}$ we assign a multiindex $\underline{d}(w)\in{\mathbb{Z}}_{+}^{N(w)}$ and an irreducible biinvariant subvariety
$\displaystyle B[w]\subset{\cal P}(\underline{d}(w))$ 
of codimension $a(w)$ with $\beta(B[w])=b(w)$.
We start the construction with $W_{0}=\{\emptyset\}$. Set $B[\emptyset]=B\subset{\cal P}(\underline{d})$, where $\underline{d}(\emptyset)=\underline{d}$, so that $a(\emptyset)=a$ and $b(\emptyset)=b$. If $b(\emptyset)=0$, then set $W_{1}=W_{2}=\cdots=W_{0}$: the procedure terminates. Assume that the subsets $W_{0}$,…, $W_{l}$ are already constructed. If for every $w\in W_{l}$ the equality $b(w)=0$ holds, then we set $W_{l+1}=W_{l+2}=\cdots=W_{l}$, terminating the procedure. Otherwise, take any word $w\in W_{l}$ with $b(w)\geqslant 1$. Now apply Theorem 3.1 to the subvariety $B[w]\subset{\cal P}(\underline{d}(w))$ (constructed at a previous step). Consider the words $w_{1}=wA$ and $w_{2}=wC_{i}$, $i\in\{0,1\}$, where $i=1$ in the case of stable reduction and $i=0$, otherwise. Furthermore,
$\displaystyle B[w_{j}]=(B[w])_{j}\subset{\cal P}(\underline{d}^{+}(w)),$ 
$j=1,2$, in the sense of notations of Theorem 3.1, so that $\underline{d}(w_{j})=(\underline{d}(w))^{+}$ and
$\displaystyle a(w_{j})=(a(w))_{j}={\rm codim}(B[w_{j}]\subset{\cal P}( \underline{d}(w_{j}))),$ 
$b(w_{1})=b(w)1$ and $b(w_{2})=b(w)i$. The inequality
$\displaystyle a(w_{1})\leqslant a(w)(2b(w)1)$ 
holds, in the case of stable reduction the same inequality ifs satisfied by the second codimension,
$\displaystyle a(w_{2})\leqslant a(w)(2b(w)1),$ 
whereas in the case of nonstable reduction the estimate
$\displaystyle a(w_{2})\leqslant a(w)b(w)$ 
holds. In any case, however, $a(w_{j})<a(w)$.
The set of words $W_{l+1}$ is obtained from $W_{l}$ by removing the word $w$ and adding the words $w_{1}$, $w_{2}$:
$\displaystyle W_{l+1}=(W_{l}\setminus\{w\})\cup\{w_{1},w_{2}\}.$ 
In particular, $\sharp W_{l+1}=\sharp W_{l}+1$. Theorem 3.1 implies that
$\displaystyle\sum_{w\in W_{l}}\mu(B[w])\leqslant\sum_{w\in W_{l+1}}\mu(B[w]).$ 
Therefore, for every $l$ we have the estimate
$\mu(B)\leqslant\sum_{w\in W_{l}}\mu(B[w])$  (8) 
Proposition 4.3.
The procedure of constructing the sets $W_{l}\subset{\cal W}$ terminates: for some $l=L$ we have $b(w)=0$ for all words $w\in W_{L}$.
Proof.
This is obvious, if, in order to construct $W_{l+1}$ we take a word $w\in W_{l}$ with the maximal value of the codimension $a(w)$, since $a(w_{j})<a(w)$ for both words $w_{j}$, $j=1,2$. However, this implies the finiteness of the procedure for any choice of the word $w\in W_{l}$: it is easy to see that the set
$\displaystyle\mathop{\bigcup}\limits_{l\in{\mathbb{Z}}_{+}}W_{l}\subset{\cal W}$ 
does not depend on which word $w\in W_{l}$ with $b(w)\geqslant 1$ is chosen at every step, and for that reason this set is finite.
One can argue in a simpler way: as we mentioned above, the length of every word does not exceed $N$. Q.E.D. for the proposition.
Set $W=W_{L}$. For any $w\in W$ we have $b(w)=0$, so that $\mu(B[w])=1$. Therefore,
$\mu(B)\leqslant\sharp W.$  (9) 
So in order to estimate from above the multiplicity $\mu(B)$, we need to estimate the cardinality of the set $W$.
4.4 An Estimate for the Cardinality of the Set of Words
We will write the words in the following way:
$\displaystyle w=\tau_{1}\cdots\tau_{K},$ 
where $\tau_{i}\in\{A,C_{0},C_{1}\}$. Now let
$\displaystyle\nu\colon\{A,C_{0},C_{1}\}\to\{A,C\}$ 
be the map from the threeletter alphabet to the twoletter one, given by the equalities $\nu(A)=A$, $\nu(C_{i})=C$, and
$\displaystyle\nu\colon w=\tau_{1}\cdots\tau_{K}\mapsto\bar{w}=\nu(\tau_{1}) \cdots\nu(\tau_{K})$ 
the corresponding map of the set of words. Now we have
Lemma 4.1.
For every $i=0,1,\ldots$ the map $\nu_{W_{i}}$ is injective. In particular, $\nu_{W}$ is injective.
Proof.
A stronger claim is true: among all words $\bar{w}=\nu(w)$, $w\in W_{i}$, no one is a left segment of another one. (In particular, no two words are equal, which means the injectivity of the map $\nu_{W_{i}}$.) The last claim is easy to show by induction. The set $W_{0}$ consists of one word, and for it the claim is trivial. Assume that we have shown it for $W_{i}$, where $i=0,\ldots,e$. If $W_{e+1}=W_{e}$, then there is nothing to prove. If $W_{e+1}\neq W_{e}$, then $W_{e+1}$ is obtained from $W_{e}$ by removing some word $w\in W_{e}$ and adding two words $w_{1}=wA$ and $w_{2}=wC_{\alpha}$, where $\alpha\in\{0,1\}$. For these words we have $\bar{w}_{1}=\bar{w}A$ and $\bar{w}_{2}=\bar{w}C$. Obviously, $\bar{w}_{1}$ and $\bar{w}_{2}$ are not left segments of each other and no word $\bar{w}^{\prime}$ for $w^{\prime}\in W_{e}\backslash\{w\}$ is not a left segment of $\bar{w}_{1}$ or $\bar{w}_{2}$, because otherwise $\bar{w}^{\prime}=\bar{w}_{1}$ or $\bar{w}_{2}$ (since $\bar{w}^{\prime}$ is not a left segment of the word $\bar{w}$ by the inductive assumption), but then $\bar{w}$ would be a left segment of the word $\bar{w}^{\prime}$, contrary to the inductive assumption. In a trivial way $\bar{w}_{1}$ and $\bar{w}_{2}$ are not left segments of the word $\bar{w}^{\prime}$, since otherwise this would have been true for $\bar{w}$ as well, contrary to the inductive assumption. Q.E.D. for the lemma. $\square$
Let $w\in W$ be a word, $w^{\prime}$ its left segment (by the construction of the set $W$ we have
$\displaystyle w^{\prime}\in\mathop{\bigcup}\limits_{l\in{\mathbb{Z}}_{+}}W_{l},$ 
since $w$ is obtained from the empty word $\emptyset$ by adding letters at the righthand end when changing from $W_{k}$ to $W_{k+1}$ for certain values $k$), and moreover, $w^{\prime}\neq w$ and $w^{\prime}\tau$ is the left segment of the word $w$ of length $w^{\prime}+1$.
Lemma 4.2.

(i)
If $\tau=A$ or $C_{1}$, then the inequality
$\displaystyle a(w^{\prime}\tau)\leqslant a(w^{\prime})(2b(w^{\prime})1)$ holds and $b(w^{\prime}\tau)=b(w^{\prime})1$.

(ii)
If $\tau=C_{0}$, then the inequality
$\displaystyle a(w^{\prime}\tau)\leqslant a(w^{\prime})b(w^{\prime})$ holds and $b(w^{\prime}\tau)=b(w^{\prime})$.
Besides, the inequality (4) implies that for every word $w\in\bigcup_{l\in{\mathbb{Z}}_{+}}W_{l}$ we have the estimate
$a(w)\geqslant b^{2}(w).$  (10) 
Example 4.1.
Let us prove Proposition 4.1 in terms of the formalism developed above. Let $b=b(\emptyset)=1$. Now for every word $w\in W_{i}$ we have the alternative: either $b(w)=0$ (and then $w\in W$), or $b(w)=1$ (and then $a(w\tau)\leqslant a(w)1$ for any letter $\tau$), so that the set $W$ is of the form
$\displaystyle A,\,\,C_{0}A,\,\,C_{0}C_{0}A,\,\,\ldots,\,\,\underbrace{C_{0}C_{ 0}\cdots C_{0}}_{k}A,\,\,\underbrace{C_{0}\cdots C_{0}}_{k}C_{1},$ 
where $k+1\leqslant a$. Therefore, $\sharp W\leqslant a+1$, as we claimed above in Sect. 4.1.
Let us come back to the general case. Recall that $a\leqslant N$.
Theorem 4.2.
The following inequality holds:
$\sharp W\leqslant 2^{b}\frac{(a\frac{b(b1)}{2})^{b}}{(b!)^{2}}.$  (11) 
Proof.
For every word $w\in W$ by construction $b(w)=0$. Since the letter $C_{0}$ does not change the value of the parameter $b$, and the letters $A$ and $C_{1}$ bring it down by 1, we may conclude that in the word $w$ there are precisely $b$ positions, occupied by the letters $A$ and $C_{1}$. Let them be the positions with numbers
$\displaystyle m_{1}+1,\,\,m_{1}+m_{2}+2,\,\,\ldots,\,\,m_{1}+m_{2}+\cdots+m_{b }+b,$ 
$m_{i}\in{\mathbb{Z}}_{+}$. By Lemma 4.2, the inequality
$\displaystyle\begin{array}[]{cclcl}0\leqslant a(w)\leqslant a&&m_{1}b&&(2b1 )\\ &&m_{2}(b1)&&(2(b1)1)\\ &&\cdots&&\\ &&m_{i}(b(i1))&&(2(b(i1))1)\\ &&\cdots&&\\ &&m_{b}&&1=\\ &&&=&ab^{2}\sum\limits^{b}_{i=1}m_{i}(b(i1))\end{array}$ 
holds, so that $(m_{1},\ldots,m_{b})$ is an arbitrary integral point in the polytope
$\displaystyle\Delta=\{x_{1}\geqslant 0,\ldots,x_{b}\geqslant 0,bx_{1}+(b1)x_{ 2}+\cdots+x_{b}\leqslant ab^{2}\}\subset{\mathbb{R}}^{b}.$ 
Therefore, even if we assume that all possible distributions of the letters $A$ and $C_{1}$ on the chosen positions are realized by words $w\in W$ (in reality this is not the case: we have a lot less words in $W$, see Remark 4.3), then the inequality
$\displaystyle\sharp W\leqslant 2^{b}\cdot\sharp(\Delta\cap{\mathbb{Z}}^{b})$ 
holds. Now let us estimate the number of integral points in $\Delta$. For that purpose, consider a larger polytope
$\displaystyle\Delta^{+}=\left\{x_{1}\geqslant 0,\ldots,x_{b}\geqslant 0,bx_{1} +\cdots+x_{b}\leqslant a\frac{b(b1)}{2}\right\}\subset{\mathbb{R}}^{b}.$ 
Obviously, $\Delta\subset\Delta^{+}$.
Lemma 4.3.
The following inequality holds:
$\displaystyle\sharp(\Delta\cap{\mathbb{Z}}^{b})\leqslant{\rm vol}(\Delta^{+}).$ 
Proof.
To every point $x=(x_{1},\ldots,x_{b})\in{\mathbb{R}}^{b}$ we correspond the unit cube
$\displaystyle\Gamma(x)=[x_{1},x_{1}+1]\times[x_{2},x_{2}+1]\times\cdots\times[ x_{b},x_{b}+1]\subset{\mathbb{R}}^{b},$ 
the vertex with the minimum value of the sum of coordinates $x_{1}+\cdots+x_{b}$ of which is the point $x$. If $x\in\Delta$, then $\Gamma(x)\subset\Delta^{+}$, since
$\displaystyle b+(b1)+\cdots+1+ab^{2}=a\frac{b(b1)}{2}.$ 
Therefore
$\displaystyle\sharp(\Delta\cap{\mathbb{Z}}^{b})=\sum_{x\in\Delta\cap{\mathbb{Z }}^{b}}{\rm vol}(\Gamma(x))={\rm vol}\left(\bigcup_{x\in\Delta\cap{\mathbb{Z}} ^{b}}\Gamma(x)\right)\leqslant{\rm vol}(\Delta^{+}),$ 
as we claimed. Q.E.D. for the lemma. $\square$
Computing the volume of the polytope $\Delta^{+}$, we complete the proof of Theorem 4.2.
4.5 Some Calculus
The inequality (11) immediately implies the estimate
$\displaystyle\mu(a)\leqslant\mathop{\max}\limits_{1\leqslant b\leqslant[\sqrt{ a}]}v_{b},$ 
where
$\displaystyle v_{b}=2^{b}\frac{(a\frac{b(b1)}{2})^{b}}{(b!)^{2}}.$ 
We have to estimate the maximum of the sequence $v_{b}$ on the set $\{1,\ldots,[\sqrt{a}]\}$ by a function that depends on the argument $a$ only. We do it in a few steps. Set
$\displaystyle u_{b}=\frac{1}{2\pi b}\left(\frac{2ab(b1)}{b^{2}}e^{2}\right)^ {b}.$ 
Lemma 4.4.
The inequality $v_{b}\leqslant u_{b}$ holds.
Proof.
Applying the Stirling formula, we write
$\displaystyle v_{b}=\frac{1}{2\pi be^{\theta/6b}}\left(\frac{2ab(b1)}{b^{2}} e^{2}\right)^{b},$ 
where $0<\theta<1$. Q.E.D. for the lemma. $\square$
Lemma 4.5.
The sequence $u_{b}$ is increasing if the following inequality holds:
$2ab(b1)\geqslant\frac{5}{2}b^{2}.$  (12) 
Proof.
Write
$\frac{u_{b+1}}{u_{b}}=\frac{1}{1+\frac{1}{b}}\frac{e^{2}}{\left(1+\frac{1}{b} \right)^{2b}}\frac{1}{\left(1+\frac{2b}{2ab(b+1)}\right)^{b}}\frac{2ab(b+1)} {(b+1)^{2}}.$  (13) 
Assume first that $b\geqslant 9$. If the numbers $a$ and $b$ satisfy the inequality $2ab(b+1)\geqslant\frac{5}{2}(b+1)^{2}$ (that is, the inequality (12) for $b+1$), then the denominator of the third factor in the right hand side can estimated from above in the following way:
$\displaystyle\left(1+\frac{2b}{2ab(b+1)}\right)^{b}\leqslant\left(1+\frac{4}{ 5}\frac{1}{b}\right)^{b}<e^{\frac{4}{5}}.$ 
The second factor in the right hand side of the inequality (13) is strictly higher than one, whereas the fourth is at least $\frac{5}{2}$. As a result, we get:
$\displaystyle\frac{u_{b+1}}{u_{b}}>\frac{9}{10}\cdot\frac{5}{2}\cdot e^{\frac {4}{5}}>1,$ 
which is what we need. For smaller values $b\leqslant 8$ the second and third factors in the right hand side of the inequality (13) can be estimated more precisely, and elementary calculations with a computer complete the proof of the lemma. $\square$
Corollary 4.1.
For $a\geqslant 17$ the value $b_{\rm max}\in\{1,\ldots,[\sqrt{a}]\}$, at which the maximum of the sequence $u_{b}$ is attained, satisfies the inequality
$\displaystyle 2ab_{\rm max}(b_{\rm max}1)\leqslant\frac{5}{3}a.$ 
Proof.
By the previous lemma, the value $b_{\rm max}$ satisfies the inequality
$\displaystyle 2ab_{\rm max}(b_{\rm max}+1)\leqslant\frac{5}{2}b_{\rm max}^{2}$ 
(otherwise, the next element of the sequence $u_{b}$ would be higher). Now elementary computations complete the proof of the corollary. $\square$
Corollary 4.2.

(i)
For $a\geqslant 17$ the following estimate holds:
$\displaystyle\sharp W\leqslant q_{b}=\frac{1}{2\pi b}\left(\frac{5a}{3b^{2}}e^ {2}\right)^{b}.$ 
(ii)
For any $a$ the following estimate holds:
$\displaystyle\sharp W\leqslant w_{b}=\frac{1}{2\pi b}\left(\frac{2a}{b^{2}}e^{ 2}\right)^{b}.$
Proof.
Both claims follow immediately from the inequality (11), taking into account Lemma 4.4 and the previous corollary. $\square$
Theorem 4.3.

(i)
For $a\geq 17$ the following estimate holds:
$\displaystyle\mu(a)\leqslant\frac{e^{2}}{2\pi[\sqrt{a}]}\left(\frac{5}{3}e^{2} \right)^{[\sqrt{a}]}.$ 
(ii)
For any $a$ the following estimate holds:
$\displaystyle\mu(a)\leqslant\frac{e^{2}}{2\pi[\sqrt{a}]}\left(2e^{2}\right)^{[ \sqrt{a}]}.$
Proof.
The arguments are identical in both cases, the only difference is which of the two claims of Corollary 4.2 is used.
Let us show part (i). Arguing as in the proof of Lemma 4.5, we conclude that the sequence $q_{b}$ is increasing. Therefore, its maximum is attained for $b=[\sqrt{a}]$. Since
$\displaystyle a<(b+1)^{2}=b^{2}+2b+1,$ 
the inequality
$\displaystyle\left(\frac{a}{b^{2}}\right)^{b}\leqslant\left(1+\frac{2}{b} \right)^{b}<e^{2},$ 
holds, which immediately implies the claim (i). The second part is shown in word for word the same way. Q.E.D. $\square$
Remark 4.3.
As we can see from the given proof, the estimate we obtained is not optimal and can be essentially improved. For $b\approx\sqrt{a}$ we have $2ab(b1)\approx a$, so that in the inequality of Theorem 4.3 the expression $(2e^{2})$ can be replaced by $e^{2}$. Furthermore, in the proof of Theorem 4.2 we took into account all possible tuples of positions $(m_{1},\ldots,m_{b})$ and all possible distributions of the letters $A$ and $C_{1}$ into $b$ positions. However, since in the set of words $\overline{W}=\nu(W)$ of the twoletter alphabet $\{A,C\}$ no word is a left segment of another word and the map $\nu\colon W\to\overline{W}$ is onetoone, for a fixed distribution of the letters $A$ and $C_{1}$ into $b$ positions, such that at least two letters $C_{1}$ follow one another, not all tuples $(m_{1},\ldots,m_{b})\in\Delta\cap{\mathbb{Z}}^{b}$ are realized, since two distinct words $w_{1}\neq w_{2}$, $\{w_{1},w_{2}\}\subset W$ can not differ only on a segment consisting of the letters $C_{0},C_{1}$. The question of finding a precise upper estimate for the numbers $\mu(a)$, even in the asymptotic sense, remains an open problem.
5 A Generalization of the Gabrielov–Khovanskii Problem
In this section we consider a generalization of the Gabrielov–Khovanskii problem: the polynomials $f_{1},\ldots,f_{N}$ are restricted onto an arbitrary (not a fixed) subvariety $R\ni o$ of codimension $l$, and for the multiplicity we take the multiplicity of zero at the point $o$ of a tuple of generic $Nl$ polynomials in the polynomial span of the tuple $\underline{f}$. In Sect. 5.1 we give a formal statement of the problem, in Sect. 5.2 its alternative setting using the Chow varieties, parameterizing effective cycles of a given degree and codimension in ${\mathbb{P}}^{N}$. In Sect. 5.3 we show that the estimates of the multiplicity of zero in the generalized problem can be obtained parallel to the estimates for the original problem (Sects. 3, 4).
5.1 Polynomial Spans and Multiplicities
We consider the affine space ${\mathbb{A}}^{N}={\mathbb{C}}^{N}_{(z_{1},\ldots,z_{N})}$ as open set embedded in the projective space ${\mathbb{P}}^{N}_{(x_{0}:x_{1}:\cdots:x_{N})}$ as the standard affine chart $\{x_{0}\neq 0\}$, that is, $z_{i}=x_{i}/x_{0}$. For an ordered multiindex $\underline{d}$ consider the space of tuples of polynomials ${\cal P}(\underline{d})$. For an arbitrary tuple $\underline{f}\in{\cal P}(\underline{d})$ we define its polynomial span
$\displaystyle[\underline{f}]=[f_{1},\ldots,f_{N}]\subset{\cal P}(\underline{d})$ 
as the smallest biinvariant set, containing the tuple $\underline{f}$. In particular, the polynomial span contains all tuples of the form $(f^{+}_{1},\ldots,f^{+}_{N})$, where
$\displaystyle f^{+}_{i}=f_{i}+\sum_{j<i}s_{i,j}(z_{*})f_{j},$ 
the polynomials $s_{i,j}(z_{*})$ run through the entire space ${\cal P}_{[0,d_{i}d_{j}],N}$ (independently of each other), so that $f^{+}_{i}\in{\cal P}_{[1,d_{i}],N}$ and $\underline{f}^{+}\in{\cal P}(\underline{d})$.
Now for an irreducible subvariety $R\subset{\mathbb{P}}^{N}$ of codimension $l\in\{0,1,\ldots,N\}$ set:

•
if $o\not\in R$, then $\mu(\underline{f};R)=0$,

•
if the closed set
$\displaystyle R\cap\{f_{1}=\cdots=f_{N}=0\}$ has an irreducible component of a positive dimension, passing through the point $o$, then $\mu(\underline{f};R)=\infty$,

•
if none of the two cases described above takes place, then
$\displaystyle\mu(\underline{f};R)=e_{\cal O}(f^{+}_{l+1},\ldots,f^{+}_{N})$ where ${\cal O}={\cal O}_{o,R}$ is the local ring at the point $o$ and $e_{\cal O}$ is the Samuel multiplicity, see Fulton ([Fulton1984], Chapter 7); $(f^{+}_{1},\ldots,f^{+}_{N})\in[f_{1},\ldots,f_{N}]$ is a general tuple.
For an arbitrary effective cycle $R=\Sigma_{j\in J}r_{j}R_{j}$ of pure codimension $l$ define $\mu(\underline{f};R)$ by linearity, setting
$\displaystyle\mu(\underline{f};R)=\sum\limits_{j\in J}r_{j}\mu(\underline{f};R _{j}),$ 
where the sum in the right hand side is $\infty$, if at least one value $\mu(\underline{f};R_{j})$ is $\infty$ (and $r_{j}\geqslant 1$). It is easy to see that if $\mu(\underline{f};R)$ is a finite nonzero number, then it is equal to the multiplicity of the point $o$ in the 0cycle
$\displaystyle(\{f^{+}_{l+1}=0\}\circ\cdots\circ\{f^{+}_{N}=0\}\circ R),$ 
where the schemetheoretic intersection is taken in a neighborhood of the point $o$.
Furthermore, set for any $\delta\geqslant 1$
$\displaystyle\mu(\underline{f},\delta)=\mathop{\sup}\limits_{{\rm deg}R=\delta }\{\mu(\underline{f};R)\},$ 
where the supremum is taken over all effective cycles $R$ on ${\mathbb{P}}^{N}$ of pure codimension $l$ and degree ${\rm deg}R=\delta$.
Now let us consider an irreducible biinvariant subvariety $B\subset{\cal P}(\underline{d})$. For an effective cycle $R$ of pure codimension $l\in\{0,\ldots,N\}$ set
$\displaystyle\mu(B;R)=\mu(\underline{f};R),$ 
where $\underline{f}\in B$ is a general tuple of polynomials. For any $\delta\geqslant 1$ set
$\displaystyle\mu(B,\delta)=\mu(\underline{f},\delta),$ 
where $\underline{f}\in B$ is a general tuple of polynomials; obviously,
$\displaystyle\mu(B,\delta)={\rm inf}\{\mu(\underline{f},\delta)\,\,\underline {f}\in B\}.$ 
Finally,
$\displaystyle\mu_{l}(a,\delta)=\mathop{\sup}\limits_{\alpha(B)\leqslant a}\{ \mu(B,\delta)\},$ 
where $\alpha(B)={\rm codim}(B\subset{\cal P}(\underline{d}))$ and the supremum is taken over all irreducible biinvariant subvarieties of codimension at most $a$ in ${\cal P}(\underline{d})$.
We emphasize that $\mu(B,\delta)$ is not
$\displaystyle\mathop{\sup}\limits_{{\rm deg}R=\delta}\{\mu(B;R)\},$ 
because the equality $\mu(B;R)=\mu(\underline{f};R)$ holds for any tuple $\underline{f}\in U_{R}$ from a nonempty Zariski open subset $U_{R}\subset B$, which depends on $R$.
The generalized Gabrielov–Khovanskii problem, considered in this section, is to compute (or estimate) the function $\mu_{l}(a,\delta)$. We will show that for $a\leqslant N$ the inductive procedure of estimating this function is totally similar to the absolute problem, considered in Sects. 2–4. The resulting estimates are linear in the degree $\delta$.
5.2 The Chow Varieties
By the symbol $\overline{\cal C}_{l,N}(\delta)$ we denote the Chow variety, parameterizing effective cycles of pure codimension $l$ and degree $\delta$ on ${\mathbb{P}}^{N}$, so that the definition of the number $\mu(\underline{f},\delta)$, given above, can be written in the following way:
$\displaystyle\mu(\underline{f},\delta)=\mathop{\sup}\limits_{\overline{\cal C} _{l,N}(\delta)\ni R}\{\mu(\underline{f};R)\}.$ 
Now let us describe an alternative definition of the numbers $\mu(B,\delta)$. Consider the sets
$\displaystyle{\cal X}_{l,N}(m,\delta)\subset{\cal P}(\underline{d})\times \overline{\cal C}_{l,N}(\delta),$ 
consisting of such tuples $(\underline{f},R)$, that
$\displaystyle\mu(\underline{f};R)\geqslant m\in{\mathbb{Z}}_{+}\cup\{\infty\}.$ 
It is easy to see that ${\cal X}_{l,N}(m,\delta)$ are closed algebraic sets. Denote by the symbol $\pi_{{\cal P}}$ the projection
$\displaystyle(\underline{f},R)\mapsto\underline{f}.$ 
By projectivity of Chow varieties we get that
$\displaystyle X_{l,N}(m,\delta)=\pi_{{\cal P}}({\cal X}_{l,N}(m,\delta)) \subset{\cal P}(\underline{d})$ 
is a closed algebraic set. Explicitly, it consists of such tuples $\underline{f}$, for which there exists and effective cycle $R\in\overline{\cal C}_{l,N}(\delta)$, satisfying the inequality $\mu(\underline{f};R)\geqslant m$.
Let $B\subset{\cal P}(\underline{d})$ be an irreducible subvariety. We define the multiplicity $\mu(B,\delta)=\mu_{l,N}(B,\delta)$ (in order to simplify the notations, we sometimes omit arguments or indices, the value of which is fixed at the moment), setting
$\displaystyle\mu(B,\delta)=\mathop{\max}\limits_{m\in{\mathbb{Z}}_{+}\cup\{ \infty\}}\{m\,\,B\subset X_{l,N}(m,\delta)\}.$ 
Explicitly: $\mu(B,\delta)=m$, if for a general tuple $\underline{f}\in B$ and any effective cycle $R\in\overline{\cal C}_{l,N}(\delta)$ the inequality
$\displaystyle\mu(\underline{f};R)\leqslant m$ 
holds, and for at least one cycle $R\in\overline{\cal C}_{l,N}(\delta)$ this inequality turns into the equality. It is clear that if $\mu_{l,N}(B,\delta)=\infty$, then for a general (and therefore, every) tuple $(\underline{f})$ the set of its zeros $Z(f_{1},\ldots,f_{N})$ has a component of positive dimension, passing through the point $o$. The converse is also true: if there is such a component, for $R$ one can take such a subvariety that ${\rm dim}(R\cap Z(f_{*}))\geqslant 1$. By the previous remark, in the notations of Sect. 2.2 the equality $X_{l,N}(\infty,\delta)=X_{\infty}$ holds and for that reason Propositions 2.1 and 2.2 give an estimate of the codimension of that set.
By construction, the sets $X_{l,N}(m,\delta)$ are biinvariant, so that in order to define and estimate the numbers $\mu_{l}(a,\delta)$ it is sufficient to consider biinvariant subsets $B\subset{\cal P}(\underline{d})$, as we did it in Sects. 2, 3. (If we replace an arbitrary subvariety $B\subset{\cal P}(\underline{d})$ by its biinvariant span $[B]\subset{\cal P}(\underline{d})$, then the codimension does not decrease and all the numbers $\mu(B;R)$, $\mu(B,\delta)$ do not change.) The further study of the numbers $\mu(B;R)$, $\mu(B,\delta)$ and $\mu_{l}(a,\delta)$ goes parallel to the constructions of Sects. 2–4, and we will only outline its main steps, paying attention to the additional arguments and modifications.
An analog of Proposition 2.3 is the following
Proposition 5.1.
If $\beta(B)=0$, then $\mu(B;R)={\rm mult}_{o}R$ for every $R\in\overline{\cal C}_{l,N}(\delta)$.
Proof.
We may assume that the subvariety $B$ is biinvariant. Let us fix an effective cycle $R$. If $o\not\in{\rm Supp}R$, then in a trivial way $\mu(B;R)={\rm mult}_{o}R=0$. So we assume that $R\subset{\mathbb{P}}^{N}$ is an irreducible subvariety containing the point $o$.
For a general tuple $\underline{f}\in B$ by assumption we have
$\displaystyle\{df_{1}(o)=\cdots=df_{N}(o)=0\}=\{0\},$ 
so that
$\displaystyle{\rm codim}\left(\{df_{1}(o)=\cdots=df_{l+1}(o)=0\}\subset{ \mathbb{C}}^{N}\right)=l+1.$ 
Since the variety $B$ is biinvariant, with every tuple $\underline{g}\in B$ it contains also the tuple $\underline{g}^{+}=(g^{+}_{1},\ldots,g^{+}_{N})$, where
$\displaystyle g_{i}^{+}=g_{i}+\sum^{i1}_{j=1}\lambda_{i,j}g_{j}$ 
for any $\lambda_{i,j}\in{\mathbb{C}}$. It follows that for a general tuple $\underline{f}\in B$ the linear form $df_{l+1}(o)$ does not vanish identically on every component of the tangent cone $T_{o}R$ (they all have codimension $l$ in ${\mathbb{C}}^{N}$). Therefore,
$\displaystyle{\rm mult}_{o}(R\circ\{f_{l+1}=0\})={\rm mult}_{o}R.$ 
This equality holds for every algebraic cycle $R$ of codimension $l$ on ${\mathbb{P}}^{N}$ and a general tuple $\underline{f}$. The cycle $(R\circ\{f_{l+1}=0\})$ has codimension $l+1$. Continuing in this way for the codimension $l+1,\ldots,N$, we complete the proof of the proposition. $\square$
Corollary 5.1.
If $\beta(B)=0$, then $\mu(B,\delta)=\delta$ for every $\delta\geqslant 1$.
Proof.
Indeed, for every effective cycle $R$ of pure codimension $l$ the inequality ${\rm mult}_{o}R\leqslant{\rm deg}R$ holds, and moreover, for the cones we have the equality. Q.E.D. for the corollary.
Assume now that $a={\rm codim}(B\subset{\cal P}(\underline{d}))\leqslant N$ and consider the procedure of reducing to the smaller dimensions, constructed in Sect. 3, and the resulting explicit estimates for the numbers $\mu_{l}(a,\delta)$, similar to those obtained for the numbers $\mu(a)$ in Sect. 4.
5.3 Reduction to the Smaller Dimensions and Explicit Estimates
in the Generalized Problem
The procedure of bringing into the standard form and subsequent splitting off a direct factor yields the following generalization of Theorem 3.1.
Theorem 5.1.
Assume that $a={\rm codim}(B\subset{\cal P}(\underline{d}))\leqslant N$ and $b=\beta(B)\geqslant 1$. Then there are irreducible biinvariant subvarieties $B_{i}\subset{\cal P}(\underline{d}^{+})$, $i=1,2$, and integers $\delta_{1},\delta_{21},\delta_{22}\in{\mathbb{Z}}_{+}$, such that $\delta=\delta_{1}+\delta_{21}+\delta_{22}$ and the inequality
$\displaystyle\mu_{l,N}(B,\delta)$  $\displaystyle\leqslant$  $\displaystyle\mu_{l,N1}(B_{1},\delta_{1})+\mu_{l,N1}(B_{2},\delta_{1})$  (14)  
$\displaystyle+\mu_{l1,N1}(B_{1},\delta_{21})+\mu_{l1,N1}(B_{2},\delta_{22}),$ 
holds, whereas the claims (ii), (iii) of Theorem 3.1 remain true.
Proof.
is almost word for word the same as the proof of Theorem 3.1. We dwell only on the necessary changes. We use the notations of Sects. 3.2–3.4.
Let $\mu_{l,N}(B,\delta)=m$. Consider a general polynomial $h_{1}h_{2}\in\pi_{\Pi}(B^{\rm st}\cap{\cal P}_{\Pi})$ and a general tuple
$\displaystyle\underline{f}=(f_{1},\ldots,f_{e1},h_{1}h_{2},f_{e+1},\ldots,f_{ N})\in B^{\rm st}\cap\pi^{1}_{\Pi}(h_{1}h_{2}).$ 
By the definition of multiplicity, there is an effective cycle $R\in\overline{\cal C}_{l,N}(\delta)$, such that $\mu(\underline{f};R)\geqslant m$. We define effective cycles $R_{1}$, $R_{21}$ and $R_{22}$ by the following conditions:

•
$R=R_{1}+R_{21}+R_{22}$,

•
the component $Q$ of the cycle $R$ is contained in $R_{1}$ if and only if $Q\not\subset H_{1}$ and $Q\not\subset H_{2}$,

•
the component $Q$ of the cycle $R$ is contained in $R_{21}$ if and only if $Q\subset H_{1}$,

•
the component $Q$ of the cycle $R$ is contained in $R_{22}$ if and only if $Q\not\subset H_{1}$, but $Q\subset H_{2}$,
Set $\delta_{1}={\rm deg}R_{1}$, $\delta_{2i}={\rm deg}R_{2i}$, $i=1,2$. Obviously, $\delta=\delta_{1}+\delta_{21}+\delta_{22}$. Consider an arbitrary irreducible component $Q$ of the cycle $R$. It is an irreducible component of precisely one of the cycles $R_{1},R_{21},R_{22}$.
If $Q\subset{\rm Supp}R_{1}$, then
$\displaystyle\mu(\underline{f};Q)\leqslant\mu(\rho_{1}(\underline{f}^{+});(Q \circ H_{1}))+\mu(\rho_{2}(\underline{f}^{+});(Q\circ H_{2})).$ 
Here $(Q\circ H_{i})$ are effective cycles of pure codimension $l$ on $H_{i}\cong{\mathbb{P}}^{N1}$.
If $Q\subset{\rm Supp}R_{21}$, then
$\displaystyle\mu(\underline{f};Q)\leqslant\mu(\rho_{1}(\underline{f}^{+});Q),$ 
where $Q$ is considered as a subvariety of codimension $(l1)$ on $H_{1}\cong{\mathbb{P}}^{N1}$; in a similar way, if $Q\subset{\rm Supp}R_{22}$, then
$\displaystyle\mu(\underline{f};Q)\leqslant\mu(\rho_{2}(\underline{f}^{+});Q),$ 
where $Q\subset H_{2}\cong{\mathbb{P}}^{N1}$ is a subvariety of codimension $(l1)$. This proves the inequality (14). The remaining part of the proof of Theorem 3.1 works as it is.
Q.E.D. for Theorem 5.1. $\square$
Remark 5.1.
If we fix an effective cycle $R\in\overline{\cal C}_{l,N}(\delta)$, then the following claim is an analog of Theorem 3.1: if $a\leqslant N$ and $\beta(B)\geqslant 1$, then there are irreducible biinvariant subvarieties $B_{i}\subset{\cal P}(\underline{d}^{+})$, $i=1,2$, and effective cycles $R_{1}\in\overline{\cal C}_{l,N1}(\delta_{1})$ and $R_{2}\in\overline{\cal C}_{l1,N1}(\delta_{2})$ of degrees $\delta_{1},\delta_{2}\in{\mathbb{Z}}_{+}$, where $\delta=\delta_{1}+\delta_{2}$, such that the inequality
$\mu(B;R)\leqslant\mu(B_{1};R_{1})+\mu(B_{2};R_{1})+\mu(B_{2};R_{2})$  (15) 
holds, and moreover, the claims (ii), (iii) of Theorem 3.1 hold. The proof is word for word the same as the proof of the previous claim, with the only difference: for a fixed cycle $R$ the general hyperplane $H_{1}$ does not contain any of its irreducible components, so that in the notations of the proof of Theorem 5.1 one can set $R_{21}=0$.
Now similar to Proposition 4.1 we get
Corollary 5.2.
For $a\leqslant N$, $\beta(B)=1$ the inequality $\mu_{l,N}(B,\delta)\leqslant(a+1)\delta$ holds.
Proof.
is given by induction on $N$. By the inequality (14) and Corollary 5.1 we have the estimate
$\displaystyle\mu_{l,N}(B,\delta)\leqslant\delta_{1}+\delta_{21}+\mu_{l,N1}(B_ {2},\delta_{1})+\mu_{l1,N1}(B_{2},\delta_{22}).$ 
If $\beta(B_{2})=0$, then $\mu_{l,N}(B,\delta)\leqslant 2\delta_{1}+\delta_{21}+\delta_{22}\leqslant 2\delta$, which what we need. If $\beta(B_{2})=1$, then by the inductive assumption
$\displaystyle\mu_{l,N}(B,\delta)\leqslant\delta_{1}+\delta_{21}+a\delta_{1}+a \delta_{22}\leqslant(a+1)\delta.$ 
Q.E.D. for the corollary. $\square$
In Sect. 4.2 we introduced the function ${\bar{\mu}}(a,b)$.
Corollary 5.3.
The inequality
$\displaystyle\mu_{l,N}(B,\delta)\leqslant{\bar{\mu}}(a,b)\delta,$ 
holds, where $a={\rm codim}(B\subset{\cal P}(\underline{d}))\leqslant N$ $b=\beta(B)$.
As an analog on the inequality (9), we have the estimate
$\mu_{l,N}(B,\delta)\leqslant(\sharp W)\delta,$  (16) 
which is obtained by repeating the arguments of Sect. 4.3 word for word, taking into account the equality $\delta=\delta_{1}+\delta_{21}+\delta_{22}$ at every step. The corresponding formal procedure is constructing irreducible biinvariant subvarieties $B[w]$, parameterized by the words of threeletter alphabet $\{A,C_{0},C_{1}\}$, and nonnegative integers $\delta_{j}(w)$, $j=0,\ldots,{\rm min}\{l,w\}$, satisfying the equality
$\displaystyle\delta=\sum^{{\rm min}\{l,w\}}_{j=0}\delta_{j}(w).$ 
As an analog of the estimate (8), we have the estimate
$\displaystyle\mu_{l,N}(B,\delta)\leqslant\sum_{w\in W_{l}}\sum^{{\rm min}\{l, w\}}_{j=0}\mu_{lj,Nw}(B[w],\delta_{j}(w)),$ 
shown by induction on $l=0,1,\ldots$. If for a word $w$ we have $\beta(B[w])=0$, then Corollary 5.1 allows us to replace the summand $\mu_{lj,Nw}(B[w],\delta_{j}(w))$ by the number $\delta_{j}(w)$. The details are left to the reader. Proof of the inequality (16) is complete.
Theorem 4.2 now implies
Theorem 5.2.
Assume that $a={\rm codim}(B\subset{\cal P}(\underline{d}))\leqslant N$ and $b=\beta(B)\geqslant 1$. Then for every $\delta\in{\mathbb{Z}}_{+}$ the following inequality holds:
$\displaystyle\mu_{l,N}(B,\delta)\leqslant 2^{b}\frac{(a\frac{b(b1)}{2})^{b}} {(b!)^{2}}\delta.$ 
More convenient estimates follow from Theorem 4.3.
Theorem 5.3.
For $a\geq 17$ the estimate
$\displaystyle\mu_{l,N}(B,\delta)\leqslant\frac{e^{2}}{2\pi[\sqrt{a}]}\left( \frac{5}{3}e^{2}\right)^{[\sqrt{a}]}\delta$ 
holds, and for every $a\geqslant 1$ the following estimate holds:
$\displaystyle\mu_{l,N}(B,\delta)\leqslant\frac{e^{2}}{2\pi[\sqrt{a}]}\left(2e^ {2}\right)^{[\sqrt{a}]}\delta.$ 
Corresponding estimates are true for the suprema $\mu_{l}(a,\delta)$ as well.