Research ContributionArnold Mathematical Journal

Received: 7 November 2014 / Accepted: 3 June 2015

Homology of Spaces of Non-Resultant Homogeneous Polynomial Systems in ${\mathbb{R}}^{2}$ and ${\mathbb{C}}^{2}$

V. A. Vassiliev Supported by Program “Leading scientific schools”, Grant No. NSh-5138.2014.1 and RFBR Grant 13-01-00383Steklov Mathematical Institute of Russian Academy of Sciences National Research University Higher School of Economics Moscow Russia

The resultant variety in the space of systems of homogeneous polynomials of some given degrees consists of such systems having non-trivial solutions. We calculate the integer cohomology groups of all spaces of non-resultant systems of polynomials ${\mathbb{R}}^{2}\to{\mathbb{R}}$, and also the rational cohomology rings of spaces of non-resultant systems and non-$m$-discriminant polynomials in ${\mathbb{C}}^{2}$.

Resultant variety, Simplicial resolution, Spectralsequence,
Configuration space, Caratheodory theorem

1 Introduction

Given $n$ natural numbers $d_{1}\geq d_{2}\geq\dots\geq d_{n}$, consider the space of all real homogeneous polynomial systems

$\left\{{\!}\begin{aligned}\displaystyle a_{1,0}x^{d_{1}}+a_{1,1}x^{d_{1}-1}y+% \dots+a_{1,{d_{1}}}y^{d_{1}}\\ \displaystyle\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\\ \displaystyle a_{n,0}x^{d_{n}}+a_{n,1}x^{d_{n}-1}y+\dots+a_{n,{d_{n}}}y^{d_{n}% }\\ \end{aligned}\right.$ (1)

in two real variables $x,y$.

We will refer to this space as ${\mathbb{R}}^{D}$, $D=\sum_{1}^{n}(d_{i}+1)$. The resultant variety $\Sigma\subset{\mathbb{R}}^{D}$ is the space of all systems having non-zero solutions. $\Sigma$ is a semialgebraic subvariety of codimension $n-1$ in ${\mathbb{R}}^{D}$.

Below we calculate the cohomology group of its complement, $H^{*}({\mathbb{R}}^{D}{\setminus}\Sigma)$. Also, we calculate the rational cohomology rings of the complex analogs ${\mathbb{C}}^{D}{\setminus}\Sigma_{\mathbb{C}}$ of all spaces ${\mathbb{R}}^{D}{\setminus}\Sigma$.

For the “affine” version of the “real” problem (concerning the space of non-resultant systems of polynomials ${\mathbb{R}}^{1}\to{\mathbb{R}}^{1}$ with leading terms $x^{d_{i}}$), see, e.g., [Vassiliev1994]; [Vassiliev1997] and [Kozlowski and Yamaguchi2000]; for the “complex” problem with $n=2$ see also [Cohen et al.1991]. A similar calculation for spaces of real homogeneous polynomials in ${\mathbb{R}}^{2}$ without zeros of multiplicity $\geq m$ was done in [Vassiliev1998].

The entire study of homology groups of spaces of non-singular (in appropriate sense) objects goes back to the Arnold’s works ([Arnold1970], [Arnold1989]), as well as the idea of using the Alexander duality in this problem.

2 Main Results

2.1 Notation

For any natural $p$, denote by $N(p)$ the sum of all numbers $d_{i}+1,$ $i=1,\dots,n,$ which are less than or equal to $p$, plus $p$ times the number of those $d_{i}$ which are equal to or greater than $p$. [In other words, $N(p)$ is the area of the part of Young diagram $(d_{1}+1,\dots,d_{n}+1)$ strictly to the left from the $(p+1)$-th column.] Let the index $\Upsilon(p)$ be equal to the number of even numbers $d_{i}\geq p$ if $p$ is even, and to the number of odd numbers $d_{i}\geq p$ if $p$ is odd. By $\tilde{H}^{*}(X)$ we denote the cohomology group reduced modulo a point. ${\overline{H}}_{*}(X)$ denotes the Borel–Moore homology group, i.e. the homology group of the complex of locally finite singular chains of $X$.

Theorem 1.

If the space ${\mathbb{R}}^{D}{\setminus}\Sigma$ is non-empty $($i.e. either $n>1$ or $d_{1}$ is even$)$, then the group $\tilde{H}^{*}({\mathbb{R}}^{D}{\setminus}\Sigma,{\mathbb{Z}})$ is equal to the direct sum of following groups:

A$)$ For any $p=1,\dots,d_{3}$,

if $\Upsilon(p)$ is even, then ${\mathbb{Z}}$ in dimension $N(p)-2p$ and ${\mathbb{Z}}$ in dimension $N(p)-2p+1$,

if $\Upsilon(p)$ is odd, then only one group ${\mathbb{Z}}_{2}$ in dimension $N(p)-2p+1$;

B$)$ If $d_{1}-d_{2}$ is odd, then an additional summand ${\mathbb{Z}}$ in dimension $D-d_{1}-d_{2}-2$. If $d_{1}-d_{2}$ is even, then an additional summand ${\mathbb{Z}}^{d_{2}-d_{3}+1}$ in dimension $D-d_{1}-d_{2}-1$ and $($if $d_{2}\neq d_{3})$ a summand ${\mathbb{Z}}^{d_{2}-d_{3}}$ in dimension $D-d_{1}-d_{2}-2$.

Example 1.

Let $n=2$ [so that part (A) in the statement of Theorem 1 is void]. If $d_{1}$ and $d_{2}$ are of the same parity, then ${\mathbb{R}}^{D}{\setminus}\Sigma$ consists of $d_{2}+1$ connected components, each of which is homotopy equivalent to a circle. For an invariant, which separates systems belonging to different components, we can take the index of the induced map of the unit circle $S^{1}\subset{\mathbb{R}}^{2}$ into ${\mathbb{R}}^{2}{\setminus}0$. This index can take all values of the same parity as $d_{1}$ and $d_{2}$ from the segment $[-d_{2},d_{2}]$. The 1-dimensional cohomology class inside any component is just the rotation number of the image of a fixed point [say, $(1,0)$] around the origin. Moreover, the images of this point under our non-resultant systems define a map ${\mathbb{R}}^{D}{\setminus}\Sigma\to{\mathbb{R}}^{2}{\setminus}0;$ it is easy to see that any fiber of this map consists of $d_{2}+1$ contractible components.

If $d_{1}$ and $d_{2}$ are of different parities, then the space ${\mathbb{R}}^{D}{\setminus}\Sigma$ has the homology of a two-point set. The invariant separating its two connected components can be calculated as the parity of the number of zeros of the odd-degree polynomial of our non-resultant system, which lie in the (well-defined) domain in ${\mathbb{RP}}^{1}$ where the even-degree polynomial is positive.

Now, let ${\mathbb{C}}^{D}$ be the space of all polynomial systems (1) with complex coefficients $a_{i,j}$, and $\Sigma_{\mathbb{C}}\subset{\mathbb{C}}^{D}$ the set of systems having solutions in ${\mathbb{C}}^{2}{\setminus}0$.

Theorem 2.

For any $n>1$, the ring $H^{*}({\mathbb{C}}^{D}{\setminus}\Sigma_{{\mathbb{C}}},{\mathbb{Q}})$ is an exterior algebra over ${\mathbb{Q}}$ with two generators of dimensions $2n-3$ and $2n-1$. Namely, these generators are the linking number with the Borel–Moore fundamental class of entire resultant variety and the pull-back of the basic cohomology class under the map ${\mathbb{C}}^{D}{\setminus}\Sigma_{\mathbb{C}}\to{\mathbb{C}}^{n}{\setminus}0$ defined by restrictions of non-resultant systems $(f_{1},\dots,f_{n})$ to the point $(1,0)$. The weight filtrations of these two generators and their product in the mixed Hodge structure of ${\mathbb{C}}^{D}{\setminus}\Sigma_{\mathbb{C}}$ are equal to $2n-2$, $2n$ and $4n-2$ respectively.

Consider also the space ${\mathbb{C}}^{d+1}$ of all complex homogeneous polynomials


and $m$-discriminant $\Sigma_{m}$ in it consisting of all polynomials vanishing on some line with multiplicity $\geq m$.

Theorem 3.

For any $m>1$ and $d\geq 2m-1$, the ring $H^{*}({\mathbb{C}}^{d+1}{\setminus}\Sigma_{m},{\mathbb{Q}})$ is isomorphic to an exterior algebra over ${\mathbb{Q}}$ with two generators of dimensions $2m-3$ and $2m-1$. The weight filtrations of these two generators and of their product are equal to $2m-2$, $2m$ and $4m-2$ respectively. For any $m>1$ and $d\in[m+1,2m-2]$, this ring is isomorphic to ${\mathbb{Q}}$ in dimensions $0,2m-3,2m-1$ and $2d-2$, and is trivial in all other dimensions; the multiplication is obviously trivial. For $d=m>1$ this ring is isomorphic to ${\mathbb{Q}}$ in dimensions 0 and $2m-3$, and is trivial in all other dimensions.

3 Some Preliminary Facts

Denote by $B(M,p)$ the configuration space of subsets of cardinality $p$ of a topological space $M$.

Lemma 1.

For any natural $p,$ there is a locally trivial fiber bundle $B(S^{1},p)\to S^{1}$ whose fiber is homeomorphic to ${\mathbb{R}}^{p-1}$. This fiber bundle is non-orientable if $p$ is even, and is orientable $($and hence trivial$)$ if $p$ is odd.

Indeed, the projection of this fiber bundle can be realised as the product of $p$ points of the unit circle in ${\mathbb{C}}^{1}$. The fiber of this bundle can be identified in terms of the universal covering ${\mathbb{R}}^{p}\to T^{p}$ with any connected component of some hyperplane $\{x_{1}+\dots+x_{p}=\mbox{const}\}$, from which all affine planes given by $x_{i}=x_{j}+2\pi k$, $i\neq j$, $k\in{\mathbb{Z}}$, are removed. Such a component is convex and hence diffeomorphic to ${\mathbb{R}}^{p-1}$. The assertion on orientability can be checked immediately. $\square$

Let us embed a manifold $M$ generically into the space ${\mathbb{R}}^{T}$ of a very large dimension, and denote by $M^{*r}$ the union of all $(r-1)$-dimensional simplices in ${\mathbb{R}}^{T}$, whose vertices lie in this embedded manifold (and the “genericity” of the embedding means that if two such simplices have a common point in ${\mathbb{R}}^{T}$, then their minimal faces containing this point coincide).

Proposition 1.

(C. Caratheodory theorem: see also [Vassiliev1997]; [Kallel and Karoui2011]) For any $r\geq 1$, the space $(S^{1})^{*r}$ is homeomorphic to $S^{2r-1}$.

Remark 1.

This homeomorphism can be realized as follows. Consider the space ${\mathbb{R}}^{2r+1}$ of all real homogeneous polynomials ${\mathbb{R}}^{2}\to{\mathbb{R}}^{1}$ of degree $2r$, the convex cone in this space consisting of everywhere non-negative polynomials, and (also convex) dual cone in the dual space $\widehat{\mathbb{R}}^{2r+1}$ consisting of linear forms taking only positive values inside the previous cone. The intersection of the boundary of this dual cone with the unit sphere in $\widehat{\mathbb{R}}^{2r+1}$ is naturally homeomorphic to $(S^{1})^{*r}$; on the other hand it is homeomorphic to the boundary of a convex $2r$-dimensional domain.

Lemma 2.

(see [Vassiliev1999], Lemma 3) For any $r>1$, the group $H_{*}((S^{2})^{*r},{\mathbb{Q}})$ is trivial in all positive dimensions. $\square$

Consider the “sign local system” $\pm{\mathbb{Q}}$ over $B({\mathbb{CP}}^{1},p)$, i.e. the local system of groups with fiber ${\mathbb{Q}}$ such that the elements of $\pi_{1}(B({\mathbb{CP}}^{1},p))$ defining odd (respectively, even) permutations of $p$ points in ${\mathbb{CP}}^{1}$ act in the fiber as multiplication by $-1$ (respectively, by 1).

Lemma 3.

(see [Vassiliev1999], Lemma 2) All Borel–Moore homology groups $\overline{H}_{i}(B({\mathbb{CP}}^{1},p);\pm{\mathbb{Q}})$ with $p\geq 1$ are trivial except

$\overline{H}_{0}(B({\mathbb{CP}}^{1},1),\pm{\mathbb{Q}})\,{\cong}\,\overline{H% }_{2}(B({\mathbb{CP}}^{1},1),\pm{\mathbb{Q}})\,{\cong}\,\overline{H}_{2}(B({% \mathbb{CP}}^{1},2),\pm{\mathbb{Q}})\,{\cong}\,{\mathbb{Q}}.$


4 Proof of Theorem 1

Following [Arnold1970], we use the Alexander duality

$\tilde{H}^{i}({\mathbb{R}}^{D}{\setminus}\Sigma)\simeq{\overline{H}}_{D-i-1}(% \Sigma).$ (2)

4.1 Simplicial Resolution of $\Sigma$

To calculate the right-hand group in (2), we construct a resolution of the space $\Sigma$. Let $\chi:{\mathbb{RP}}^{1}\to{\mathbb{R}}^{T}$ be a generic embedding, $T\gg d_{1}$. For any system $\Phi=(f_{1},\dots,f_{n})\in\Sigma$ not equal identically to zero, consider the simplex $\Delta(\Phi)$ in ${\mathbb{R}}^{T}$ spanned by the images $\chi(x_{i})$ of all points $x_{i}\in{\mathbb{RP}}^{1}$ corresponding to all lines, on which the system $f$ has a common root. (The maximal possible number of such lines is obviously equal to $d_{1}.$)

Furthermore, consider a subset in the direct product ${\mathbb{R}}^{D}\times{\mathbb{R}}^{T}$, namely, the union of all simplices of the form $\Phi\times\Delta(\Phi),$ $\Phi\in\Sigma{\setminus}0$. This union is not closed: the set of its limit points not belonging to it is the product of the point $0\in{\mathbb{R}}^{D}$ (corresponding to the zero system) and the union of all simplices in ${\mathbb{R}}^{T}$ spanned by the images of no more than $d_{1}$ different points of the line ${\mathbb{RP}}^{1}.$ By the Caratheodory theorem, the latter union is homeomorphic to the sphere $S^{2d_{1}-1}.$ We can assume that our embedding $\chi:{\mathbb{RP}}^{1}\to{\mathbb{R}}^{T}$ is algebraic, and hence this sphere is semialgebraic. Take a generic $2d_{1}$-dimensional semialgebraic disc in ${\mathbb{R}}^{T}$ bounded by this sphere (e.g., the union of segments connecting the points of this sphere with a generic point in ${\mathbb{R}}^{T}$), and add the product of the point $0\in{\mathbb{R}}^{D}$ and this disc to the previous union of simplices $\Phi\times\Delta(\Phi)\subset{\mathbb{R}}^{D}\times{\mathbb{R}}^{T}$. The resulting closed subset in ${\mathbb{R}}^{D}\times{\mathbb{R}}^{T}$ will be denoted by $\sigma$ and called a simplicial resolution of $\Sigma$.

Lemma 4.

The obvious projection $\sigma\to\Sigma$ (induced by the projection of ${\mathbb{R}}^{D}\times{\mathbb{R}}^{T}$ onto the first factor) is proper, and the induced map between one-point compactifications of these spaces is a homotopy equivalence.

This follows easily from the fact that this projection is a stratified map of semialgebraic spaces, and the preimage of any point of $\Sigma$ is contractible: see [Vassiliev1994]; [Vassiliev1997]. $\square$

So, we can (and will) calculate the group ${\overline{H}}_{*}(\sigma)$ instead of ${\overline{H}}_{*}(\Sigma)$.

Remark 2.

There is a different construction of a simplicial resolution of $\Sigma$ in terms of “Hilbert schemes”. Namely, let $I_{p}$ be the space of all ideals of codimension $p$ in the space of smooth functions ${\mathbb{RP}}^{1}\to{\mathbb{R}}^{1}$ equipped with the natural “Grassmannian” topology. It is easy to see that $I_{p}$ is homeomorphic to the $p$-th symmetric power $S^{p}({\mathbb{RP}}^{1})=({\mathbb{RP}}^{1})^{p}/S(p){;}$ in particular, it contains the configuration space $B({\mathbb{RP}}^{1},p)$ as an open dense subset. Consider the disjoint union of these $d_{1}$ spaces $I_{1},\dots,I_{d_{1}}$ augmented with the one-point set $I_{\infty}$ symbolizing the zero ideal. The incidence of ideals makes this union a partially ordered set. Consider the continuous order complex $\Xi_{d_{1}}$ of this poset, i.e. the subset in the join $I_{1}*\dots*I_{d_{1}}*I_{\infty}$ consisting of simplices, whose all vertices are incident to one another. For any polynomial system $\Phi=(f_{1},\dots,f_{n})\in{\mathbb{R}}^{D}$, denote by $\Xi(\Phi)$ the subcomplex in $\Xi_{d_{1}}$ consisting of all simplices, whose all vertices correspond to ideals containing all polynomials $f_{1},\dots,f_{n}$. The simplicial resolution $\tilde{\sigma}\subset\Sigma\times\Xi_{d_{1}}$ is defined as the union of simplices $\Phi\times\Xi(\Phi)$ over all $\Phi\in\Sigma$.

This construction is homotopy equivalent to the previous one. In particular, the Caratheodory theorem has the following version (see [Kallel and Karoui2011]): the continuous order complex of the poset of all ideals of codimension $\leq r$ in the space of functions $S^{1}\to{\mathbb{R}}^{1}$ is homotopy equivalent to $S^{2r-1}$.

However, this construction is less convenient for our practical calculations than the one described above and used previously in [Vassiliev1994]; [Vassiliev1999] [and extended to some more complicated situations in [Gorinov2005]].

The space $\sigma$ has a natural increasing filtration $F_{1}\subset\dots\subset F_{d_{1}+1}=\sigma$: its term $F_{p},$ $p\leq d_{1},$ is the closure of the union of all simplices of the form $\Phi\times\Delta(\Phi)$ over all polynomial systems $\Phi$ having no more than $p$ lines of common zeros. Alternatively, it can be described as the union of all no more than $(p-1)$-dimensional faces of all simplices $\Phi\times\Delta(\Phi)$ over all systems $\Phi\in\Sigma{\setminus}0$, completed with all no more than $(p-1)$-dimensional simplices spanning some $\leq p$ points of the manifold $\{0\}\times\chi(\mathbb{RP}^{1})$.

Lemma 5.

For any $p=1,\ldots,d_{1},$ the term $F_{p}{\setminus}F_{p-1}$ of our filtration is the space of a locally trivial fiber bundle over the configuration space $B({\mathbb{RP}}^{1},p),$ with fibers equal to the direct product of a $(p-1)$-dimensional open simplex and a $(D-N(p))$-dimensional real space. The corresponding bundle of open simplices is orientable if and only if $p$ is odd $($i.e. exactly when the base configuration space is orientable$)$, and the bundle of $(D-N(p))$-dimensional spaces is orientable if and only if the index $\Upsilon(p)$ is even.

The last term $F_{d_{1}+1}{\setminus}F_{d_{1}}$ of this filtration is homeomorphic to an open $2d_{1}$-dimensional disc.

Indeed, to any configuration $(x_{1},\ldots,x_{p})\in B({\mathbb{RP}}^{1},p),$ $p\leq d_{1}$, there corresponds the direct product of the interior part of the simplex in ${\mathbb{R}}^{T}$ spanned by the images $\chi(x_{i})$ of points of this configuration, and the subspace of ${\mathbb{R}}^{D}$ consisting of polynomial systems that have solutions on corresponding $p$ lines in ${\mathbb{R}}^{2}.$ The codimension of the latter subspace is equal exactly to $N(p)$. The assertion concerning the orientations can be checked in a straightforward way. The description of $F_{d_{1}+1}{\setminus}F_{d_{1}}$ follows immediately from the construction and the Caratheodory theorem. $\square$

Consider the spectral sequence $E_{p,q}^{r},$ calculating the group ${\overline{H}}_{*}(\Sigma)$ and generated by this filtration. Its term $E_{p,q}^{1}$ is canonically isomorphic to the group ${\overline{H}}_{p+q}(F_{p}{\setminus}F_{p-1}).$ By Lemma 5, its column $E_{p,*}^{1},$ $p\leq d_{1},$ is as follows. If $\Upsilon(p)$ is even, then this column contains exactly two non-trivial terms $E_{p,q}^{1}$, both isomorphic to ${\mathbb{Z}}$, for $q$ equal to $D-N(p)+p-1$ and $D-N(p)+p-2$. If $\Upsilon(p)$ is odd, then this column contains only one non-trivial term $E_{p,q}^{1}$ isomorphic to ${\mathbb{Z}}_{2}$, for $q=D-N(p)+p-2$. Finally, the column $E^{1}_{d_{1}+1,*}$ contains only one non-trivial element $E^{1}_{d_{1}+1,d_{1}-1}\,{\cong}\,{\mathbb{Z}}$.

Figure 1: $E^{1}$ for $n=1$, $d_{1}$ even (left) and $n=1$, $d_{1}$ odd (right)

Before calculating the differentials and further terms $E^{r}$, $r>1$, let us consider several basic examples.

4.2 The Case $n=1$

If our system consists of only one polynomial of degree $d_{1}$, then the term $E^{1}$ of our spectral sequence looks as in Fig. 1; in particular, all non-trivial groups $E_{p,q}^{1}$ lie in two rows $q=d_{1}$ and $q=d_{1}-1$.

Lemma 6.

If $n=1$, then in both cases of even or odd $d_{1}$, all possible horizontal differentials $\partial_{1}:E_{p,d_{1}-1}^{1}\to E_{p-1,d_{1}-1}^{1}$ of the form ${\mathbb{Z}}\to{\mathbb{Z}}_{2}$, $p=d_{1}+1,d_{1}-1,d_{1}-3,\dots$ are epimorphisms, and all differentials $\partial_{2}:E_{p,d_{1}-1}^{2}\to E_{p-2,d_{1}}^{2}$ of the form ${\mathbb{Z}}\to{\mathbb{Z}}$, $p=d_{1}+1,d_{1}-1,d_{1}-3,\dots$ are isomorphisms. In particular, the unique surviving term $E_{p,q}^{3}$ for the “even” spectral sequence is $E_{1,d_{1}-1}^{3}\,{\cong}\,{\mathbb{Z}}$, and for the “odd” one it is $E_{2,d_{1}-1}^{3}\,{\cong}\,{\mathbb{Z}}$.

Indeed, in both cases we know the answer. In the “odd” case, the discriminant coincides with entire ${\mathbb{R}}^{D}={\mathbb{R}}^{d_{1}+1}$. In the “even” case, its complement consists of two contractible components, so that ${\overline{H}}_{*}(\Sigma)={\mathbb{Z}}$ in dimension $d_{1}$ and is trivial in all other dimensions. Therefore, all terms $E_{p,q}$ with $p+q$ not equal to $d_{1}+1$ (respectively, to $d_{1}$) in the odd- (respectively, even-) dimensional case should die at some stage; this is possible only if all assertions of our lemma hold. $\square$

4.3 The Case $n=2$

There are two very different situations depending on the parity of $d_{1}-d_{2}$. In Fig. 2, we demonstrate these situations in two particular cases: $(d_{1},d_{2})=(6,3)$ and $(7,3)$. However, the general situation is essentially the same; namely, the following is true.

Figure 2: $E^{1}$ for $n=2$, $d_{1}-d_{2}$ odd (left) and $n=2$, $d_{1}-d_{2}$ even (right)

If $n=2$ and $d_{1}-d_{2}$ is odd, then all indices $\Upsilon(p)$, $p=1,\dots,d_{2}+1$, are odd, and hence all non-trivial groups $E_{p,q}^{1}$ with such $p$ lie on the line $\{p+q=d_{1}+d_{2}\}$ only and are equal to ${\mathbb{Z}}_{2}$.

If $n=2$ and $d_{1}-d_{2}$ is even, then all indices $\Upsilon(p)$, $p=1,\dots,d_{2}+1$, are even, and hence all non-trivial groups $E_{p,q}^{1}$ with such $p$ lie on two lines $\{p+q=d_{1}+d_{2}\}$, $\{p+q=d_{1}+d_{2}+1\}$, and are all equal to ${\mathbb{Z}}$.

In both cases, all groups $E_{p,q}^{1}$ with $p>d_{2}$ are the same as in the case $n=1$ with the same $d_{1}$. Moreover, the differentials $\partial_{1}$ and $\partial_{2}$ between these groups are also the same as for $n=1$; therefore, all of these groups die at $E^{3}$ except for $E_{d_{2}+1,d_{1}-1}^{3}\,{\cong}\,{\mathbb{Z}}$ for even $d_{1}-d_{2}$, and $E_{d_{2}+2,d_{1}-1}^{3}\,{\cong}\,{\mathbb{Z}}$ for odd $d_{1}-d_{2}$.

In the case of even $d_{1}-d_{2}$, all other differentials between the groups $E_{p,q}^{r}$ are trivial, because otherwise the group $\tilde{H}^{0}({\mathbb{R}}^{D}{\setminus}\Sigma)$ would be smaller than ${\mathbb{Z}}^{d_{2}}$, in contradiction to $d_{2}+1$ different components of this space indicated in Example 1.

On the contrary, if $d_{1}-d_{2}$ is odd, then all the differentials $d_{r}:E_{d_{2}+2,d_{1}-1}^{r}\to E_{d_{2}+2-r,d_{1}-2+r}^{r}$, $r=1,\dots,d_{1}-d_{2}+1$, are epimorphic just because the integer cohomology group of the topological space ${\mathbb{R}}^{D}{\setminus}\Sigma$ cannot have non-trivial torsion subgroup in dimension 1. Therefore, the unique nontrivial group $E_{p,q}^{\infty}$ in this case is $E_{d_{2}+2,d_{1}-1}^{\infty}\,{\cong}\,{\mathbb{Z}}$.

This proves Theorem 1 for $n=2$.

4.4 The General Case

Now suppose that our systems (1) consist of $n\geq 3$ polynomials. Let again $\sigma$ be the simplicial resolution of the corresponding resultant variety constructed in Sect. 4.1, and $\sigma^{\prime}$ be the simplicial resolution of the resultant variety for $n=2$ and the same $d_{1}$ and $d_{2}$. The parts $\sigma{\setminus}F_{d_{3}}(\sigma)$ and $\sigma^{\prime}{\setminus}F_{d_{3}}(\sigma^{\prime})$ of these resolutions are canonically homeomorphic to one another as filtered spaces. In particular, $E_{p,q}^{1}(\sigma)=E_{p,q}^{1}(\sigma^{\prime})$ if $p>d_{3}$, and $E_{p,q}^{r}(\sigma)=E_{p,q}^{r}$ if $p\geq d_{3}+r$. All non-trivial terms $E_{p,q}^{r}(\sigma)$ with $p\leq d_{3}$ are placed in such a way that no non-trivial differentials $\partial_{r}$ can act between these terms, as well as no differentials can act to these terms from the cells $E_{p,q}^{r}$ with $p>d_{3}$, which have survived the differentials between these cells described in the previous subsection.

Therefore, the final term $E_{p,q}^{\infty}(\sigma)$ coincides with $E_{p,q}^{1}(\sigma)$ in the domain $\{p\leq d_{3}\}$, and coincides with the term $E_{p,q}^{\infty}(\sigma^{\prime})$ of the truncated spectral sequence calculating the Borel–Moore homology of $\sigma^{\prime}{\setminus}F_{d_{3}}(\sigma^{\prime})$ in the domain $\{p>d_{3}\}$. This completes the proof of Theorem 1. $\square$

5 Proof of Theorem 2

The simplicial resolution $\sigma_{{\mathbb{C}}}$ of $\Sigma_{{\mathbb{C}}}$ appears in the same way as its real analog $\sigma$ in the previous section. It also has a natural filtration $F_{1}\subset\dots\subset F_{d_{1}+1}=\sigma_{{\mathbb{C}}}$. For $p\in[1,d_{1}]$, its term $F_{p}{\setminus}F_{p-1}$ is fibered over the configuration space $B({\mathbb{CP}}^{1},p)$; its fiber over a configuration $(x_{1},\dots x_{p})$ is equal to the product of the space ${\mathbb{C}}^{D-N(p)}$ (consisting of all complex systems (1) vanishing at all lines corresponding to the points of this configuration) and the $(p-1)$-dimensional simplex whose vertices correspond to the points of the configuration. In particular, our spectral sequence calculating rational Borel-Moore homology of $\sigma_{{\mathbb{C}}}$ has $E^{1}_{p,q}\,{\cong}\,{\overline{H}}_{q-2(D-N(p))+1}(B({\mathbb{CP}}^{1},p);% \pm{\mathbb{Q}})$ for such $p$. By Lemma 3, only the following such groups are non-trivial: $E^{1}_{1,2(D-n)-1}\,{\cong}\,{\mathbb{Q}}$, $E^{1}_{1,2(D-n)+1}\,{\cong}\,{\mathbb{Q}}$, and (if $d_{1}>1$) $E^{1}_{2,2(D-2n)+1}\,{\cong}\,{\mathbb{Q}}$.

The last term $F_{d_{1}+1}{\setminus}F_{d_{1}}$ is homeomorphic to the cone over the $d_{1}$-th self-join $({\mathbb{CP}}^{1})^{*d_{1}}$ with the base of this cone removed (as it belongs to $F_{d_{1}}$). Therefore, by Lemma 2, the column $E^{1}_{d_{1}+1,*}$ is trivial if $d_{1}>1$, and contains a unique non-trivial group $E^{1}_{2,1}\,{\cong}\,{\mathbb{Q}}$ if $d_{1}=1$.

So, in any case, the first sheet $E^{1}$ of our spectral sequence has only three non-trivial terms $E^{1}_{1,2(D-n)-1}$, $E^{1}_{1,2(D-n)+1}$, and $E^{1}_{2,2(D-2n)+1}$, all of which are isomorphic to ${\mathbb{Q}}$. The differentials in it are obviously trivial; therefore, the group ${\overline{H}}_{*}(\sigma)$ has three non-trivial terms in dimensions $2(D-n)$, $2(D-n)+2$, and $2(D-2n)+3$. By Alexander duality in the space ${\mathbb{C}}^{D}$, this gives us three groups $\tilde{H}^{2n-3}\,{\cong}\,{\mathbb{Q}}$, $\tilde{H}^{2n-1}\,{\cong}\,{\mathbb{Q}}$, and $\tilde{H}^{4n-4}\,{\cong}\,{\mathbb{Q}}$, and zero in all other dimensions.

All assertions of Theorem 2 concerning the ring structure, realization of cohomology classes, and the weight filtration are well-known or obvious in the case $d_{1}=1$ (when $D=2n$ and ${\mathbb{C}}^{2n}{\setminus}\Sigma_{\mathbb{C}}$ is the space of pairs of linearly independent vectors in ${\mathbb{C}}^{n}$, and is homotopy equivalent to the Stiefel manifold $V_{2}({\mathbb{C}}^{n})$). The general case can be deduced from this one by the map $P:{\mathbb{C}}^{2n}{\setminus}\Sigma_{\mathbb{C}}\to{\mathbb{C}}^{D}{\setminus% }\Sigma_{\mathbb{C}}$ sending any collection of linear functions $(f_{1},\dots,f_{n})$ to $(f_{1}^{d_{1}},\dots,f_{n}^{d_{n}})$. Indeed, the realization of $(2n-1)$-dimensional classes follows from the commutative diagram

$\require{AMScd}\begin{CD} {\mathbb{C}}^{2n}{\setminus}\Sigma_{\mathbb{C}}@>P>>{\mathbb{C}}^{D}{\setminus}\Sigma_{\mathbb{C}}\\ @VVV @VVV\\ {\mathbb{C}}^{n}{\setminus}0 @>>>{\mathbb{C}}^{n}{\setminus}0\end{CD},$

where the lower horizontal arrow is defined by


and induces an isomorphism of $(2n-1)$-dimensional rational homology groups. The assertion on the realization of $(2n-3)$-dimensional classes is obvious. The statements on the multiplication and the weight filtration follow from the naturality of these structures. $\Box$

6 Proof of Theorem 3

The additive part of this theorem can be proved in almost the same way as that of Theorem 1: see [Vassiliev1998]. In particular, we construct a simplicial resolution $\sigma_{m}$ of the $m$-discriminant variety $\Sigma_{m}$. It has a natural filtration $\Phi_{1}\subset\dots\subset\Phi_{[d/m]}\subset\Phi_{[d/m]+1}=\sigma_{m}$. The term $\Phi_{p}{\setminus}\Phi_{p-1}$, $p\leq[d/m]$, of this filtration is the space of a fiber bundle with the base $B(\mathbb{CP}^{1},p)$. Its fiber over the collection of points $(z_{1},\dots,z_{p})\subset\mathbb{CP}^{1}$ is the product of an open $(p-1)$-dimensional simplex whose vertices are related with these $p$ points, and the subspace of codimension $mp$ in ${\mathbb{C}}^{d+1}$ consisting of all polynomials having $m$-fold zeros on the corresponding $p$ lines. The term $\Phi_{[d/m]+1}{\setminus}\Phi_{[d/m]}$ appears from the zero polynomial and is the cone over the space $(\mathbb{CP}^{1})^{*[d/m]}$ with the base of this cone removed. The term $E^{1}$ of the corresponding spectral sequence can be calculated immediately with the help of Lemmas 2 and 3. Its shape implies that all further differentials of the spectral sequence are trivial, with unique exception in the case $d=m$, when all non-zero (isomorphic to ${\mathbb{Q}}$) groups of $E^{1}$ are $E^{1}_{1,3}$, $E^{1}_{1,1}$, and $E^{1}_{2,1}$. In this case, the differential $\partial_{1}:E^{1}_{2,1}\to E^{1}_{1,1}$ is an isomorphism, because the zero section of the tautological bundle over ${\mathbb{CP}}^{1}$ defines a non-zero element of the 2-dimensional Borel–Moore homology group of the space of this bundle. Therefore, the only surviving term is $E^{2}_{1,3}\,{\cong}\,{\mathbb{Q}}$; by Alexander duality, it gives us a $(2m-3)$-dimensional cohomology class.

The remaining statements of Theorem 3 are based on the following comparison lemma. Consider the map $J:{\mathbb{C}}^{d+1}\to{\mathbb{C}}^{D}$, $D=m(d+2-m),$ sending any homogeneous polynomial ${\mathbb{C}}^{2}\to{\mathbb{C}}^{1}$ of degree $d$ to the collection of all its partial derivatives of order $m-1$.

Lemma 7.

For any $d\geq m>1$, $\Sigma_{m}=J^{-1}(\Sigma_{\mathbb{C}})$. For any $d\geq 2m-1$, the induced map of cohomology groups, $J^{*}:H^{*}({\mathbb{C}}^{D}{\setminus}\Sigma_{\mathbb{C}},{\mathbb{Q}})\to H^% {*}({\mathbb{C}}^{d+1}{\setminus}\Sigma_{m},{\mathbb{Q}}),$ is an isomorphism.

This is a standard comparison theorem of our spectral sequences: see especially Section IV.7 in [Vassiliev1994]; [Vassiliev1997]. $\square$

Now the assertions of Theorem 3 on the multiplication and weight filtrations follow from the similar assertions of Theorem 2 by the naturality of these structures. $\square$

I thank O. A. Malinovskaya for a useful discussion, and also the referee for suggesting to include the assertions on Hodge structure in Theorems 2 and 3.


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