Arnold Mathematical Journal

, Volume 1, Issue 3, pp 233–242

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

• V. A. Vassiliev
Research Contribution

## Abstract

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 $${\mathbb R}^2 \rightarrow {\mathbb R}$$, and also the rational cohomology rings of spaces of non-resultant systems and non-m-discriminant polynomials in $${\mathbb C}^2$$.

## Keywords

Resultant variety Simplicial resolution Spectral sequence  Configuration space Caratheodory theorem

## 1 Introduction

Given n natural numbers $$d_1 \ge d_2 \ge \dots \ge d_n$$, consider the space of all real homogeneous polynomial systems
\begin{aligned} \left\{ \begin{aligned} a_{1,0} x^{d_1} + a_{1,1} x^{d_1-1}y + \dots + a_{1,{d_1}}y^{d_1} \\ \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \\ 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. \end{aligned}
(1)
in two real variables xy.

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 $${\mathbb R}^1 \rightarrow {\mathbb R}^1$$ with leading terms $$x^{d_i}$$), see, e.g., Vassiliev (1994, 1997) and Kozlowski and Yamaguchi (2000); 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 $$\ge m$$ was done in Vassiliev (1998).

The entire study of homology groups of spaces of non-singular (in appropriate sense) objects goes back to the Arnold’s works (1970, 1989), 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 $$d_i \ge p$$ if p is even, and to the number of odd numbers $$d_i \ge p$$ 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 $$d_2 \ne d_3)$$ 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 $${\mathbb R}^D {\setminus } \Sigma \rightarrow {\mathbb R}^2 {\setminus } 0;$$ 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 $${\mathbb C}^D {\setminus } \Sigma _{\mathbb C}\rightarrow {\mathbb C}^n {\setminus } 0$$ 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
\begin{aligned} a_0 x^{d} + a_1 x^{d-1}y + \dots + a_dy^{d}{,} \end{aligned}
and m-discriminant $$\Sigma _m$$ in it consisting of all polynomials vanishing on some line with multiplicity $$\ge m$$.

### Theorem 3

For any $$m>1$$ and $$d \ge 2m-1$$, 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(Mp) 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 $$B(S^1,p) \rightarrow S^1$$ 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 $${\mathbb R}^p \rightarrow T^p$$ with any connected component of some hyperplane $$\{x_1 + \dots + x_p = \text{ const }\}$$, from which all affine planes given by $$x_i = x_j + 2\pi k$$, $$i \ne j$$, $$k \in {\mathbb Z}$$, 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 Vassiliev 1997; Kallel and Karoui 2011) For any $$r \ge 1$$, 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 $${\mathbb R}^2 \rightarrow {\mathbb R}^1$$ 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 Vassiliev 1999, 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 Vassiliev 1999, Lemma 2) All Borel–Moore homology groups $$\overline{H}_i(B({\mathbb {CP}}^1,p);\pm {\mathbb Q})$$ with $$p \ge 1$$ are trivial except
\begin{aligned} \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}. \end{aligned}
$$\square$$

## 4 Proof of Theorem 1

Following Arnold (1970), we use the Alexander duality
\begin{aligned} \tilde{H}^i({\mathbb R}^D {\setminus }\Sigma ) \simeq {\overline{H}} _{D-i-1}(\Sigma ). \end{aligned}
(2)

### 4.1 Simplicial Resolution of $$\Sigma$$

To calculate the right-hand group in (2), we construct a resolution of the space $$\Sigma$$. Let $$\chi : {\mathbb {RP}}^1 \rightarrow {\mathbb R}^T$$ 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 $$\chi : {\mathbb {RP}}^1 \rightarrow {\mathbb R}^T$$ 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 $$\sigma \rightarrow \Sigma$$ (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 Vassiliev (1994, 1997). $$\square$$

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 $${\mathbb {RP}}^1 \rightarrow {\mathbb R}^1$$ 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 Karoui (2011)): the continuous order complex of the poset of all ideals of codimension $$\le r$$ in the space of functions $$S^1 \rightarrow {\mathbb R}^1$$ 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 Vassiliev (1994, 1999) [and extended to some more complicated situations in Gorinov (2005)].

The space $$\sigma$$ has a natural increasing filtration $$F_1 \subset \dots \subset F_{d_1+1} = \sigma$$: its term $$F_p,$$ $$p \le d_1,$$ 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 $$\le p$$ 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 $$(x_1, \ldots , x_p) \in B({\mathbb {RP}}^1,p),$$ $$p \le d_1$$, 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 $$E_{p,*}^1,$$ $$p \le d_1,$$ 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 $$E^1_{d_1+1,d_1-1} \,{\cong }\, {\mathbb Z}$$.

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 $$\partial _1: E_{p,d_1-1}^1 \rightarrow E_{p-1,d_1-1}^1$$ of the form $${\mathbb Z}\rightarrow {\mathbb Z}_2$$, $$p=d_1+1, d_1-1, d_1-3, \dots$$ are epimorphisms, and all differentials $$\partial _2: E_{p,d_1-1}^2 \rightarrow E_{p-2,d_1}^2$$ of the form $${\mathbb Z}\rightarrow {\mathbb Z}$$, $$p=d_1+1, d_1-1, d_1-3, \dots$$ 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.

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 $$d_r:E_{d_2+2,d_1-1}^r \rightarrow E_{d_2+2-r, d_1-2+r}^r$$, $$r=1, \dots , d_1-d_2+1$$, 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 $$n\ge 3$$ polynomials. Let again $$\sigma$$ be the simplicial resolution of the corresponding resultant variety constructed in Sect. 4.1, and $$\sigma '$$ 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 $$\sigma ' {\setminus } F_{d_3}(\sigma ')$$ of these resolutions are canonically homeomorphic to one another as filtered spaces. In particular, $$E_{p,q}^1(\sigma ) = E_{p,q}^1(\sigma ')$$ if $$p > d_3$$, and $$E_{p,q}^r(\sigma ) = E_{p,q}^r$$ if $$p \ge d_3+r$$. All non-trivial terms $$E_{p,q}^r(\sigma )$$ with $$p \le d_3$$ 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 $$\{p \le d_3\}$$, and coincides with the term $$E_{p,q}^\infty (\sigma ')$$ of the truncated spectral sequence calculating the Borel–Moore homology of $$\sigma ' {\setminus } F_{d_3}(\sigma ')$$ 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 $$P: {\mathbb C}^{2n} {\setminus } \Sigma _{\mathbb C}\rightarrow {\mathbb C}^D {\setminus } \Sigma _{\mathbb C}$$ 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
\begin{aligned} \begin{array}{lcl} {\mathbb C}^{2n} {\setminus } \Sigma _{\mathbb C}&{} \mathop {\longrightarrow }\limits ^{P}&{} {\mathbb C}^D {\setminus } \Sigma _{\mathbb C}\\ \ \downarrow &{} &{} \ \downarrow \\ {\mathbb C}^n {\setminus } 0 &{} \longrightarrow &{} {\mathbb C}^n {\setminus } 0 \end{array}, \end{aligned}
where the lower horizontal arrow is defined by
\begin{aligned} (z_1, \dots , z_n) \mapsto (z_1^{d_1}, \dots , z_n^{d_n}) \end{aligned}
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 Vassiliev (1998). 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 $$\Phi _p {\setminus } \Phi _{p-1}$$, $$p \le [d/m]$$, 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 $$\partial _1: E^1_{2,1} \rightarrow E^1_{1,1}$$ 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 $$J: {\mathbb C}^{d+1} \rightarrow {\mathbb C}^D$$, $$D=m(d+2-m),$$ sending any homogeneous polynomial $${\mathbb C}^2 \rightarrow {\mathbb C}^1$$ of degree d to the collection of all its partial derivatives of order $$m-1$$.

### Lemma 7

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

This is a standard comparison theorem of our spectral sequences: see especially Section IV.7 in Vassiliev (1994, 1997). $$\square$$

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. $$\square$$

## Notes

### Acknowledgments

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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