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

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 $X$.

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 ${\mathbb{C}}^{2}{\setminus}0$.

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 $\geq m$.

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 $\Sigma$.

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 ${\overline{H}}_{*}(\Sigma)$.

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 $\{0\}\times\chi(\mathbb{RP}^{1})$.

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 $E^{1}_{d_{1}+1,d_{1}-1}\,{\cong}\,{\mathbb{Z}}$.

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

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 $q=d_{1}-1$.

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. $\square$

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 $d_{r}:E_{d_{2}+2,d_{1}-1}^{r}\to E_{d_{2}+2-r,d_{1}-2+r}^{r}$, $r=1,\dots,d_{1}-d_{2}+1$, 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 $n=2$.

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. $\square$