The Gabrielov–Khovanskii Problem for Polynomials

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

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

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

Obviously, if $\mu(a)=m$, then for the codimension of the set ${\cal F}(m)$ we get the estimate

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.

Now let us consider the behaviour of the numbers $\mu(a)$ for growing values of the number of variables $N$.

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:

Therefore, $\mu(b^{2})\geqslant 2^{b}$. The function $\mu(a)$ is obviously non-decreasing, so that we finally get the inequality $\mu(a)\geqslant 2^{[\sqrt{a}]}$ for $a\leqslant N^{2}$.

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

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

The numbers $\mu^{*}(a)$ are defined for all $a\leqslant{\rm dim}{\cal F}$.

where ${\cal O}={\cal O}_{o,X}$, is well defined (when $X$ is non-singular 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

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

in word for word the same way as it was done in Sect. 1.1 and set the generalized Gabrielov–Khovanskii problem.

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.

Let us describe briefly one important application of the Gabrielov–Khovanskii problem for polynomials. In birational geometry of higher-dimensional 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 non-singular (for simplicity) point. Consider a subvariety $Y\subset X$. One needs to estimate the ratio of the multiplicity to the degree:

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 $c=3/(N-1)$, see [Pukhlikov2013b], Chapter 3). Of course, one can always take $c=1$, but this estimate, as a rule, is insufficient, especially in higher-dimensional 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

a polynomial, such that $f|_{X}\equiv 0$, where $q_{i}(z)$ are homogeneous of degree $i$. Now the polynomial

has degree $\leqslant i$, however

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

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

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 higher-dimensional 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

an effective 1-cycle (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

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 1-cycle $C$, 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.

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

for instance, ${\cal P}_{[1,d],N}$ is the space of (non-homogeneous) 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

where $2\leqslant d_{1}\leqslant\cdots\leqslant d_{N}$, and set

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 above-mentioned group $G_{1}$, act two more groups of transformations, which we will now define. The group $G_{21}$ consists of transformations of the form

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 $d\in{\cal D}$

so that $\sum_{d\in{\cal D}}n_{d}=N$. Now the group $G_{22}$ is defined as the matrix group (realized by block-wise diagonal matrices, where the blocks of the size $n_{d}\times n_{d}$ are ordered by increasing of the integers $d$)

acting on the tuples $(f_{*})\in{\cal P}(\underline{d})$ by linear transformations of the form

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 bi-invariant, if it is invariant with respect to the action of both groups $G_{1}$ and $G_{2}$.

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 bi-invariant. For an arbitrary irreducible subvariety $B\subset{\cal P}(\underline{d})$ set

so that $\mu(B)=\mu(\underline{f})$ for a general tuple $\underline{f}\in B$. Furthermore, set

that is, the maximum is taken over all irreducible subvarieties of codimension $a$. If $\langle B\rangle$ is the bi-invariant span of the subvariety $B$, that is, the smallest bi-invariant 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 bi-invariant 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

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

for some line $L\ni o$.

Set $X^{+}_{\infty}=\overline{X_{\infty}\backslash X_{\rm line}}$.

Below we obtain estimates from above for $\mu(a)$, which are close to optimal ones, for the values $a\leqslant N$.

Let us consider the construction of the Example 1.6 more formally.

Let $B\subset{\cal P}(\underline{d})$ be an irreducible bi-invariant subvariety of codimension $a$ and $\beta(B)=b\geqslant 1$; as we have seen, $a\geqslant b^{2}$. Consider the subset

$\sharp I=b$, such that for a general tuples $\underline{f}\in B$

Because of the bi-invariance 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

On the open set $B^{\circ}$ the map of bringing into the standard form $\sigma\colon B^{\circ}\to\widetilde{\cal P}(\underline{d})$,

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

Set $B^{\rm st}=\overline{\sigma(B^{\circ})}\subset\widetilde{\cal P}(\underline{d})$. Setting for $j=1,\ldots,N$

we see that the fibre of general position $\sigma\colon B^{\circ}\to B^{\rm st}$ is of dimension

(this inequality becomes an equality if $I=\{N-b+1,\ldots,N\}$). Taking into account that

we conclude that

Now let us represent the space $\widetilde{\cal P}(\underline{d})$ as the direct product of the spaces

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

(Here we identify the fibre $\pi^{-1}(\pi(\underline{f}))$ and the space ${\cal P}_{[2,d_{e}],N}$.) Therefore, the equality

holds, whereas the right hand side does not exceed $a-b^{2}-\gamma$, so that the estimate

holds. Before studying the just constructed map $\pi_{B}$ in full generality, let us consider some simple examples.

Now let us consider the general case. Let $\underline{d}^{+}=(d_{1},\ldots,d_{e-1},d_{e+1},\ldots,d_{N})$ be the truncated tuple of degrees and

the corresponding space of tuples of $(N-1)$ polynomials in $(N-1)$ 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 $\beta(B)=b$.

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 bi-invariance of $B$, we show that the intersection is non-empty 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

$i=1,2$. The subvarieties $\{h_{i}=0\}$ are hyperplanes and can be identified with ${\mathbb{C}}^{N-1}$. 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 $(N-1)$-uples of polynomials in $(N-1)$ variables. (It turns out that one of the linear forms, say $h_{1}(z)$, can be pre-selected.) The main work in the proof is to estimate the parameters of the new subvarieties $B_{i}$.

Now we proceed to the rigorous argument.

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 $H_{1}=\{h_{1}=0\}$:

so that $\beta(B_{1})=b-1$. 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

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

contains also all tuples $(\widetilde{f}_{1},\ldots,\widetilde{f}_{e-1},\widetilde{f}_{e+1},\ldots, \widetilde{f}_{N})$, where $\widetilde{f}_{j}=f_{j}$ for $j\not\in I$ and

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

Taking into account that $\varepsilon(e)\leqslant\sharp\{j\not\in I\}=N-b$, we obtain the estimate (6). This completes the proof of the claim (ii).

Let us show the claim (iii). For a general tuple of polynomials

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: $\beta(B_{2})=b-1$ 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

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

with uniquely determined coefficients $\lambda_{i}$. Therefore, on the Zariski open subset

(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:

The natural ambient space for the right-hand side is

(up to a permutation of the direct factors), and the codimension of the closed subset

with respect to that ambient space does not exceed

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

The variety $B_{2}$ with every tuple

contains all tuples $(\widetilde{g}_{i}\,|\,i\neq e)$, where $\widetilde{g}_{j}=g_{j}$ for $j\not\in I$, $j\neq m$ and

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

Since obviously $\sharp\{j\in I\backslash\{e\}\,|\,j>m\}\leqslant b-1$ and $\varepsilon(e)\leqslant N-b$, 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 $\gamma=N-1$. 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.$\square$

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 $k=\left[\frac{\gamma}{N}\right]+1\geqslant 2$.

Let ${\cal P}(\underline{d})$ be an irreducible bi-invariant subvariety of codimension $a\leqslant N$.

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

setting $\overline{\mu}(a,0)\equiv 1$, $\overline{\mu}(a,1)\equiv a+1$, for $a<b(b+1)$

for $a\geqslant b(b+1)$

Theorem 3.1 immediately implies

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 $a$.

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 $N^{\prime}<N$.

The symbol $B$ stands for an irreducible bi-invariant subvariety of codimension $a$ with $\beta(B)=b$. Let us consider the three-letter 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

does not exceed $N$. Set $N(w)=N-|w|\in{\mathbb{Z}}_{+}$. 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 multi-index $\underline{d}(w)\in{\mathbb{Z}}_{+}^{N(w)}$ and an irreducible bi-invariant subvariety

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,

$j=1,2$, in the sense of notations of Theorem 3.1, so that $\underline{d}(w_{j})=(\underline{d}(w))^{+}$ and

$b(w_{1})=b(w)-1$ and $b(w_{2})=b(w)-i$. The inequality

holds, in the case of stable reduction the same inequality ifs satisfied by the second codimension,

whereas in the case of non-stable reduction the estimate

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 $w_{1}$, $w_{2}$:

In particular, $\sharp W_{l+1}=\sharp W_{l}+1$. Theorem 3.1 implies that

Therefore, for every $l$ we have the estimate

We will write the words in the following way:

where $\tau_{i}\in\{A,C_{0},C_{1}\}$. Now let

be the map from the three-letter alphabet to the two-letter one, given by the equalities $\nu(A)=A$, $\nu(C_{i})=C$, and

the corresponding map of the set of words. Now we have

Let $w\in W$ be a word, $w^{\prime}$ its left segment (by the construction of the set $W$ we have

since $w$ is obtained from the empty word $\emptyset$ by adding letters at the right-hand 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 $|w^{\prime}|+1$.

Besides, the inequality (4) implies that for every word $w\in\bigcup_{l\in{\mathbb{Z}}_{+}}W_{l}$ we have the estimate

Let us come back to the general case. Recall that $a\leqslant N$.

Computing the volume of the polytope $\Delta^{+}$, we complete the proof of Theorem 4.2.

The inequality (11) immediately implies the estimate

where

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

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 multi-index $\underline{d}$ 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

as the smallest bi-invariant set, containing the tuple $\underline{f}$. In particular, the polynomial span contains all tuples of the form $(f^{+}_{1},\ldots,f^{+}_{N})$, where

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

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 non-zero number, then it is equal to the multiplicity of the point $o$ in the 0-cycle

where the scheme-theoretic intersection is taken in a neighborhood of the point $o$.

Furthermore, set for any $\delta\geqslant 1$

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 bi-invariant subvariety $B\subset{\cal P}(\underline{d})$. For an effective cycle $R$ of pure codimension $l\in\{0,\ldots,N\}$ set

where $\underline{f}\in B$ is a general tuple of polynomials. For any $\delta\geqslant 1$ set

where $\underline{f}\in B$ is a general tuple of polynomials; obviously,

Finally,

where $\alpha(B)={\rm codim}(B\subset{\cal P}(\underline{d}))$ and the supremum is taken over all irreducible bi-invariant subvarieties of codimension at most $a$ in ${\cal P}(\underline{d})$.

We emphasize that $\mu(B,\delta)$ is not

because the equality $\mu(B;R)=\mu(\underline{f};R)$ holds for any tuple $\underline{f}\in U_{R}$ from a non-empty 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 $\delta$.

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:

Now let us describe an alternative definition of the numbers $\mu(B,\delta)$. Consider the sets

consisting of such tuples $(\underline{f},R)$, that

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

By projectivity of Chow varieties we get that

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

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

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 bi-invariant, so that in order to define and estimate the numbers $\mu_{l}(a,\delta)$ it is sufficient to consider bi-invariant 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 bi-invariant 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

The procedure of bringing into the standard form and subsequent splitting off a direct factor yields the following generalization of Theorem 3.1.