Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 39 (1998), No. 1, 53-74.

On Associated Discriminants for Polynomials in one Variable

To Vladinir Igorevich Arnold on the occasion of his 60-th birthday

A. Gorodentsev ^{1} B. Shapiro

Independent University of Moscow e-mail: gorod@ium.ips.ras.ru Department of Mathematics, University of Stockholm, S-10691, Sweden e-mail: shapiro@matematik.su.se

Abstract. In this note we introduce a family Σ_{i}, i = 0, …, n − 2 of discriminants in the space P_{n} of polynomials of degree n in one variable and study some of their algebraic and topological properties following [1], [10] and [6]. The discriminant Σ_{i} consists of all polynomials p such that some nontrivial linear combination α_{0}p + α_{1}p′+⋯+α_{i}p^{(i)} has a zero of multiplicity greater or equal i+2. In particular, using the inversion of differential operators with constant coefficients (which induces the nonlinear involution on P_{n}) we obtain the algebraic isomorphism of Σ_{i} and Σ_{n−2−i} for all i.

0. Introduction

It is well known that every solution to a finite nonperiodic Toda lattice

ȧ_{i} = a_{i}(b_{i+1} − b_{i}) , ḃ_{i} = a_{i} − a_{i−1}

(i = 0, ..., i ; a_{i}, b_{i} ∈ ℝ ; a_{0}a_{1}...a_{n−1} ≠ 0 ; a_{−1} = a_{n} = 0),

can be presented as

a_{i}(x) = \frac{Δ_{i−1}^{y}(x)Δ_{i+1}^{y}(x)}{(Δ_{i}^{y}(x))^{2}} , b_{i}(x) = \frac{d}{dx} ln (\frac{Δ_{i}^{y}(x)}{Δ_{i−1}^{y}(x)}) ,

1This project was started while the first author was visiting the University of Stockholm supported by the grant R-RA 01599-307 of the Swedish Natural Sciences Research Council.

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} 1998 Heldermann Verlag

54 A. Gorodentsev; B. Shapiro: On associated discriminats for polynomials in one variable

where Δ_{i}^{y}(x) is the i-th principal minor of the Hankel matrix

H_{y}(x) = (\begin{matrix}y(x) & … & y^{(n)}(x) \\ y′(x) & … & y^{(n+1)}(x) \\ ⋮ & ... & ⋮ \\ y^{(n)}(x) & … & y^{(2n)}(x) \end{matrix}) ,

and y(x) is some solution to the linear ordinary differential equation with constant coefficients of order n+1 determined by the Toda lattice, see e.g. [2]. Therefore singularities of solutions correspond to the zeros of the determinants of Δ_{i}^{y}(x). They are also intrinsically related with Schubert calculus, see [4],[7].

The subset of solutions for which the i-th principal minor Δ_{i}^{y}(x) has a multiple zero for some x is a hypersurface in the space of all solutions. The union of these hypersurfaces separates the space of (real) solutions into domains of solutions with different qualitative behavior.

A similar situation occurs both in the theory of linear Hamiltonian systems and linear ordinary differential equations with one essential difference that instead of the space of solutions one has to consider the space of fundamental systems. Vanishing of principal minors is related, for example, to the index of the trajectory of a Lagrange subspace. Again the space of fundamental solutions contains certain discriminants formed by all fundamental solutions for which at least one of principal minors vanishes with multiplicity ≥ 2. The study of the stratification of the space of (fundamental) solutions coming from the union of these discriminants is an important open problem even for linear Hamiltonian systems and ordinary differential equations with constant coefficients.

The present paper is an attempt to study some properties of the above discriminants in the space of polynomials, i.e. in the space of solutions to the simplest equation x^{(n+1)} = 0. The paper is organized as follows. Section 1 contains some general information on induced and associated discriminants. In Section 2 we present the corresponding Sylvester formula for associated discriminants. Section 3 contains various algebro-topological information, i.e. duality induced by a special nonlinear involution of the space of monic polynomials, resolution of singularities and a natural stratification. Finally, in Section 4 we calculate the cohomology with compact supports for a specific example.

Acknowledgements. The authors are sincerely grateful to M. Gekhtman and M. Shapiro who attracted our attention to the singularities of solutions of nonperiodic Toda lattices and mentioned the reference [8]. The authors want to acknowledge the hospitality, nice research atmosphere and financial support provided by the Department of Mathematics of the University of Stockholm during the beginning and by the Max-Planck-Institute during the final stage of this project.

1. Induced discriminants Σ̃_{i} and generalized multiple zeros

1.1. Notations. Given a function y(x) and a positive integer i let us consider Δ_{i}^{y}(x) the determinant of the i-th principal subminor of the above matrix H_{y}(x). Zeros of Δ_{i}^{y}(x) are called (i − 1)-generalized roots of y. (We shift the index for the sake of convenience.)

Consider now a family F(x, λ_{1}, …, λ_{k}) of functions in one variable x. The i-th induced discriminant of the family F is the set Σ̃_{i}(F) of values of parameters λ_{1}, …, λ_{k} for which Δ_{i+1}^{F} (considered as a function in x only) has a multiple zero.

The i-th associated discriminant of the family F is the set Σ_{i}(F) of values of parameters λ_{1}, …, λ_{k} for which Δ_{i} satisfies the following condition. There exists a nontrivial linear combination α_{0}F + α_{1}F′ + ⋯ + α_{i}F^{(i)} which has a zero of multiplicity i + 2 at some point x.

The relation between the induced discriminants and the associated discriminants is described by the following proposition.

1.2. Proposition. The i-th induced discriminant Σ̃_{i}(F), i ≥ 1, of the family F coincides with the union of two associated discriminants Σ_{i−1}(F) ∪ Σ_{i}(F) while Σ̃_{0}(F) = Σ_{0}(F).

Proof. First of all, Σ̃_{0} = Σ_{0} by definition. The proof of the fact Σ̃_{i}(F) = Σ_{i−1}(F) ∪ Σ_{i}(F) for all i > 1 is based on manipulations with the subminors in the following Hankel matrix of the family F(x, λ_{1}, … λ_{k})

H_{F}(x, λ_{1}, …, λ_{n}) = (\begin{matrix}F, & F′, & …, & F^{(i+2)} \\ F′, & F′′, & …, & F^{(i+3)} \\ ⋮ & ⋮ & ... & ⋮ \\ F^{(i+2)}, & F^{(i+3)}, & …, & F^{(2i+4)} \end{matrix}) ,

where F^{(j)} denotes the jth derivative of F w.r.t. x. Let us denote by H_{m_{1},...,m_{q}}^{l_{1},...,l_{p}} the submatrix in H_{F} formed by the rows with numbers l_{1}, …, l_{p} and the columns with the numbers m_{1}, …, m_{q}. By det_{m_{1},...,m_{q}}^{l_{1},...,l_{p}} we denote the determinant of H_{m_{1},...,m_{q}}^{l_{1},...,l_{p}}.

Σ̃_{i} consists of all ϕ ∈ F such that the determinant Δ_{i+1}^{ϕ}(x) = det_{1,...,i+1}^{1,...,i+1} has a multiple zero as a function in x, i.e. there exists x such that Δ_{i+1}^{F} = (Δ_{i+1}^{F})′ = 0. Differentiating Δ_{i+1}^{F} w.r.t. x one gets that (Δ^{F})′ = det_{1,...,i,i+2}^{1,...,i+1}. Thus ϕ ∈ Σ̃_{i}(F) if and only if there exists x such that det_{1,...,i+1}^{1,...,i+1} = det_{1,...,i,i+2}^{1,...,i+1} = 0.

At the same time ϕ ∈ Σ_{i−1}(F) if ∃x such that the submatrix H_{1,...,i}^{1,...,i+1} has the rank < i and ϕ ∈ Σ_{i}(F) if ∃x such that the submatrix H_{1,...,i+1}^{1,...,i+2} or equivalently H_{1,...,i+2}^{1,...,i+1}c has the rank < i + 1.

Obviously, the union Σ_{i−1}(F) ∪ Σ_{i}(F) is contained in Σ̃_{i}(F). Indeed, if ∃x such that H_{1,...,i}^{1,...,i+1} has the rank < i then for the same x one gets det_{1,...,i+1}^{1,...,i+1} = det_{1,...,i,i+2}^{1,...,i+1} = 0. Analogously, if ∃x such that the rank of H_{1,...,i+2}^{1,...,i+1} is less than i + 1 then all its (i + 1) × (i + 1)- determinants vanish and, in particular, det_{1,...,i+1}^{1,...,i+1} = det_{1,...,i,i+2}^{1,...,i+1} = 0.

Let us prove the opposite inclusion. Given ϕ and x such that det_{1,...,i+1}^{1,...,i+1} = det_{1,...,i,i+2}^{1,...,i+1} = 0 we get two cases. If for this ϕ and x the submatrix H_{1,...,i}^{1,...,i+1} has the rank < i then ϕ belongs to Σ_{i−2}(F) by definition. At the same time if the rank of H_{1,...,i}^{1,...,i+1} equals i then the system det_{1,...,i+1}^{1,...,i+1} = det_{1,...,i,i+2}^{1,...,i+1} = 0 is exactly equivalent to rk H_{1,...,i+2}^{1,...,i+1} < i + 1. □

1.3. Generalized discriminants and generalized roots for polynomials. In what follows we will study the generalized discriminants only for the families of polynomials over the field ℂ. Let P_{n} denote the set of all monic polynomials of degree n

p(x) = x^{n} + λ_{1}x^{n−1} + ⋯ + λ_{n}, λ_{i} ∈ ℂ .

From now we will omit the indication of the family if F = P_{n}.

For P_{n} only the first n determinants Δ_{m}, m = 1, …, n are functions depending on parameters λ_{1}, …, λ_{n} and Δ_{n+1} = (n+1)!^{n+1}. Thus the only nontrivial induced discriminants of P_{n} are Σ̃_{0}, …, Σ̃_{n−1}.

Let us denote by Σ_{i} the standard associated discriminant Σ_{i}(P_{n}). Σ_{0} is the usual discriminant (also called the swallowtail), i.e. the set of polynomials with multiple zeros and the 0-generalized multiple roots are the usual multiple roots. One has Σ̃_{0} = Σ_{0}, Σ̃_{n−1} = Σ_{n−2}, Σ̃_{i} = Σ_{i−1} ∩ Σ_{i} for i = 1, …, n − 2.

In the last part of this section we show that any polynomial has a finite number of igeneralized roots counted with multiplicities and therefore a finite number of pairwise different multiple roots. This result can be also interpreted as the estimation of the complexity of the selfintersection of the discriminant Σ_{i}.

1.3.1. Proposition. The number of i-generalized roots (counted with multiplicities) of any polynomial p of degree n is equal to (i + 1)(n − i). Thus the number of multiple i-generalized roots of any polynomial of degree n does not exceed \frac{(i+1)(n−i)}{2}.

Proof. One can easily see that for any p(x) ∈ P_{n} the degree of Δ_{i+1}(p) as a function of x is equal to (i + 1)(n − i). Let us now consider in the space P_{(i+1)(n−i)} the following 2 subsets, namely, the n-dimensional image Δ_{i+1}(P_{n}) and the usual discriminant Σ_{0} consisting of all polynomials with multiple zeros. It is not hard to show that Δ_{i+1}(P_{n}) ⊄ Σ_{0}. Thus a typical polynomial in Δ_{i+1}(P_{n}) has exactly (i + 1)(n − i) pairwise different i-generalized zeros. The estimation of the number of pairwise different multiple roots follows. □

1.3.2. Remark. To calculate the exact value of the maximal number of pairwise different multiple i-generalized roots for polynomials of degree n is apparently a very nontrivial problem for all i > 0. (For i = 0 the obvious answer is [\frac{n}{2}]).

The authors have made some calculations for i = 1 and small n. Recall that in this case the upper bound for the maximal number ♯_{max}(i, n) of multiple 1-generalized roots equals n − 1.

For i = 1 and n = 2, 3, 4, 5, 6 the number ♯_{max}(i, n) equals 1, 1, 3, 4, 4 resp. For the interesting cases n = 4 and n = 5 the corresponding polynomials p with the maximal number of pairwise different 1-generalized roots are of the form x^{4} + ax and x^{5} + ax resp.

2. Sylvester formula for Σ_{i}

2.1. Projective closure Σ̂_{i} of Σ_{i}. In order to write an explicit equation for the hypersurface Σ_{i} let us consider a more general homogeneous problem. Let ℙ_{1} = ℙ(U) be the projective line obtained by projectivization of a 2-dimensional vector space U with a base {e_{0}, e_{1}} and let {t_{0}, t_{1}} be the dual base of U^{∗}. We denote by V_{n} the space of all homogeneous forms of degree n in (t_{0} : t_{1}):

V_{n} = { p(t) = ∑_{ν=0}^{n} λ_{ν}t_{0}^{ν}t_{1}^{n−ν} | λ_{ν} ∈ ℂ } .

Let us consider the linear operator D : V_{n} → V_{n} sending an element p to t_{0} \frac{∂p}{∂t_{1}} and denote by D_{i} ⊂ End_{ℂ}(V_{n}) the subspace of all operators of the form

Ψ(D) = ψ_{0} + ψ_{1}D + ⋯ + ψ_{i}D^{i}, ψ_{j} ∈ ℂ .

2.1.1. Lemma. Projective closure Σ̂_{i} ⊂ ℙ_{n} = ℙ(V_{n}) of the surface Σ_{i} consists of all forms p such that Ψp has a zero of multiplicity ≥ i + 2 at some point u ∈ ℙ_{1} for some Ψ ∈ ℙ(D_{i}).

Proof. The standard affine chart {t_{0} = 1} on ℙ_{1} = ℙ(U) is an affine line 𝔸_{1} with the coordinate x = t_{1}/t_{0} and the standard affine chart {λ_{0} = 1} on ℙ_{n} = ℙ(V_{n}) coincides with the affine space P_{n} of all monic polynomials of order n. The operator D = \frac{d}{dx} on this space is induced

by the operator D = t_{0} \frac{∂}{∂t_{1}} on the space V_{n}. It remains to note that for any p ∈ P_{n} and

any nontrivial i-tuple α_{0}, α_{1}, …, α_{i} the polynomial α_{0}p + α_{1}p′ + ⋯ + α_{i}p^{(i)} has no zeros of multiplisity ≥ i + 2 at infinity, because its degree is greater or equal n − i. Therefore the affine restriction of the hypersurface Σ̂_{i} coincides with our original discriminant Σ_{i}. □

2.2. Derivation of determinantal formula. In order to write an explicit homogeneous equation defining the hypersurface Σ̂_{i}, consider the product

ℙ_{i} × ℙ_{1} = ℙ(D_{i}) × ℙ(U)

and denote by S(d_{1}, d_{2}) the space of bihomogeneous forms of bidegree (d_{1}, d_{2}) on this product. In other words, S(d_{1}, d_{2}) is the space of forms

F (Ψ, t) = F (ψ_{0}, …, ψ_{i}; t_{0}, t_{1})

on two groups of variables Ψ = (ψ_{0}, …, ψ_{i}) and t = (t_{0}, t_{1}), which are homogeneous of degree d_{1} in Ψ and homogeneous of degree d_{2} in t. We associate to each p ∈ V_{n} the collection of i + 2 forms F_{p}^{0}, F_{p}^{1}, …, F_{p}^{i+1} ∈ S(1, n − i − 1) defined by

F_{p}^{α}(Ψ, t) = (\frac{∂}{∂t_{0}} )^{α} (\frac{∂}{∂t_{1}} )^{i+1−α} Ψp(t) , α = 0, 1, …, i + 1.

2.2.1. Lemma. A form Ψp has a zero of multiplicity ≥ i + 2 at a point t ∈ ℙ_{1} if and only if all i + 2 forms F_{p}^{ν} vanish at the pair (Ψ, t) ∈ ℙ_{i} × ℙ_{1}.

Proof. This follows immediately from Taylors formula for Ψp. □

2.2.2. Resultant hypersurface. It is shown in [6] (ch. 13, §2) that all collections of i + 2 forms on ℙ_{i} × ℙ_{1}, which have a common zero, form an irreducible hypersurface in the projective space of all collections of i+2 forms on ℙ_{i}×ℙ_{1} of the same bidegree. An irreducible equation of this hypersurface is called the resultant of i + 2 forms of a given bidegree.

In our case the condition that i + 2 forms F_{p}^{0}, … F_{p}^{i+1} have a common zero is equivalent (see [6], p.439) to the fact that the ideal (F_{p}^{0}, … F_{p}^{i+1}) generated by these forms does not contain the subspace S(2, 2n − 2i − 3). More precisely, if we consider the linear operator

∂_{p} : \underbrace{S(1, n − i − 2) ⊕ ⋯ ⊕ S(1, n − i − 2)}_{i+2} ⟶ S(2, 2n − 2i − 3)

induced by multiplication by our forms F_{p}^{0}, … F_{p}^{i+1}, i. e.

∂_{p} : (G_{0}, G_{1}, …, G_{i+1}) ↦ ∑_{α=0}^{i+1} G_{α}F_{p}^{α} ,

then the forms F_{p}^{ν} have a common zero on ℙ_{i}×ℙ_{1} if and only if this operator is not epimorphic. Since both spaces S(1, n − i − 2)^{⊕i+2} and S(2, 2n − 2i − 3) have the same dimension equal to (i + 1)(i + 2)(n − i − 1) the last condition takes the form det (∂_{p}) = 0 as soon as we choose some bases in both spaces. So, we get

2.2.3. Proposition. The projective closure Σ̂_{i} ⊂ ℙ_{n} = ℙ(V_{n}) of the discriminant Σ_{i} ⊂ P_{n} is an irreducible hypersurface of degree (i + 1)(i + 2)(n − i − 1).

Proof. As we have seen above, the condition that for given p ∈ V_{n} there exists a nontrivial operator Ψ ∈ ℙ(D_{i}) such that Ψp has a root of multiplicity ≥ i + 2 at some point on ℙ_{1} is equivalent to the determinantal condition det (∂_{p}) = 0, where ∂_{p} is the square matrix of the size (i + 1)(i + 2)(n − i − 1) the entries of which are equal to some coefficients of p multiplied by appropriate constants. □

2.3. Some examples of Sylvester formula. In order to write down the matrix ∂_{p}, let us fix the standard monomial bases in all spaces. Namely, in the space

\underbrace{S(1, n − i − 2) ⊕ ⋯ ⊕ S(1, n − i − 2)}_{i+2}

we fix the basis {ψ_{β}^{(α)}t_{0}^{γ}t_{1}^{n−i−2−γ}}, where α = 0, …, i + 1 enumerates the direct summands, γ = 0, …, n − i − 2 indicates the power of t, and β = 0, …, i enumerates the basic vectors of D_{i}^{∗} giving the coefficients of Ψ = ∑ ψ_{β}D^{β}.

Similarly, in the space S(2, 2n − 2i − 3) we fix the basis {ψ_{j}ψ_{k}t_{0}^{l}t_{1}^{2n−2i−3−l}}. (Note that ψ_{j}ψ_{k} and ψ_{k}ψ_{j} are two different notations for the same basic vectors; they are convenient and we hope this will not lead to any confusion in the following formulas).

After some efforts one gets:

∂_{p}(ψ_{β}^{(α)}t_{0}^{γ}t_{1}^{n−i−2−γ}) = ∑_{μ,ν} ζ_{αβγ}^{μν} · ψ_{β}ψ_{α+μ}t_{0}^{γ+ν}t_{1}^{2n−2i−3−(γ+ν)} ,

where the summation is taken over all μ, ν such that

\begin{cases}0 ≤ ν ≤ n − i − 1 \\ 0 ≤ γ + ν ≤ 2n − 2i − 3 \\ −(n − ν) ≤ μ ≤ ν \\ 0 ≤ α + μ ≤ i \end{cases} .

The matrix elements ζ_{αβγ}^{μν} are given by

ζ_{αβγ}^{μν} = \underbrace{(ν + 1)(ν + 2) … (ν + α)}_{α} · \underbrace{(n − ν − i)(n − ν − i + 1) … (n − ν + μ)}_{i+1+μ} · λ_{ν−μ} ,

where the factors in both underbraced groups are consequently increasing positive integers.

2.3.1. Example. The usual discriminant corresponds to the case i = 0. In this case for a given p = λ_{0}t_{1}^{n} + λ_{1}t_{0}t_{1}^{n−1} + ⋯ + λ_{n}t_{0}^{n} ∈ V_{n} we have to construct two forms

F_{p}^{0} = \frac{∂p}{∂t_{0}} = λ_{1}t_{1}^{n−1} + 2λ_{2}t_{0}t_{1}^{n−2} + ⋯ + nλ_{n}t_{0}^{n−1}

F_{p}^{1} = \frac{∂p}{∂t_{1}} = nλ_{0}t_{1}^{n−1} + (n − 1)λ_{1}t_{0}t_{1}^{n−2} + ⋯ + λ_{n−1}t_{0}^{n−1}

and consider the linear operator

∂_{p} : V_{n−2} ⊕ V_{n−2} → V_{2n−3}

sending (G_{0}, G_{1}) to G_{0}F_{p}^{0} + G_{1}F_{p}^{1}.

The above bases specialize to the standard monomial bases t_{0}^{α}t_{1}^{β} of the spaces in question and the matrix of ∂_{p} gives the classical Sylvester representation for the usual discriminant of a homogeneous form p. For example, we get the well-known formula

det ∂_{p} = det (\begin{matrix}λ_{1} & 2λ_{2} & 3λ_{3} & 0 \\ 0 & λ_{1} & 2λ_{2} & 3λ_{3} \\ 3λ_{0} & 2λ_{1} & λ_{2} & 0 \\ 0 & 3λ_{0} & 2λ_{1} & λ_{2} \end{matrix}) = 3(4λ_{1}^{3}λ_{3} + 4λ_{0}λ_{2}^{3} + 27λ_{0}^{2}λ_{3}^{2} − λ_{1}^{2}λ_{2}^{2} − 18λ_{0}λ_{1}λ_{2}λ_{3})

for n = 3 and

det ∂_{p} = det (\begin{matrix}λ_{1} & 2λ_{2} & 3λ_{3} & 4λ_{4} & 0 & 0 \\ 0 & λ_{1} & 2λ_{2} & 3λ_{3} & 4λ_{4} & 0 \\ 0 & 0 & λ_{1} & 2λ_{2} & 3λ_{3} & 4λ_{4} \\ 4λ_{0} & 3λ_{1} & 2λ_{2} & λ_{3} & 0 & 0 \\ 0 & 4λ_{0} & 3λ_{1} & 2λ_{2} & λ_{3} & 0 \\ 0 & 0 & 4λ_{0} & 3λ_{1} & 2λ_{2} & λ_{3} \end{matrix}) =

= 16(−λ_{1}^{2}λ_{2}^{2}λ_{3}^{2} − 256λ_{4}^{3}λ_{0}^{3} + 4λ_{1}^{2}λ_{2}^{3}λ_{4} + 27λ_{0}^{2}λ_{3}^{4} + 4λ_{1}^{3}λ_{3}^{3} + 27λ_{1}^{4}λ_{4}^{2} + 6λ_{1}^{2}λ_{3}^{2}λ_{4}λ_{0} + 192λ_{1}λ_{3}λ_{4}^{2}λ_{0}^{2} − 18λ_{1}λ_{3}^{3}λ_{0}λ_{2}−18λ_{1}^{3}λ_{3}λ_{4}λ_{2}+80λ_{1}λ_{3}λ_{4}λ_{0}λ_{2}^{2}−144λ_{4}λ_{0}^{2}λ_{3}^{2}λ_{2}−144λ_{1}^{2}λ_{4}^{2}λ_{0}λ_{2}+128λ_{4}^{2}λ_{0}^{2}λ_{2}^{2}+4λ_{2}^{3}λ_{3}^{2}λ_{0}− 166λ_{2}^{4}λ_{4}λ_{0}) for n = 4.

2.3.2. Example. In another extremal case i = n − 2 for a given p = ∑ λ_{ν}t_{0}^{ν}t_{1}^{n−ν} we have to construct n bilinear forms

F^{α}(ψ, t) = ∑_{j=0}^{n−2} ψ_{j}(c_{j0}^{α}t_{0} + c_{j1}^{α}t_{1}) , α = 0, 1, …, n − 1 .

Coefficient in parenthesis is equal to (\frac{∂}{∂t_{0}} )^{α} (\frac{∂}{∂t_{1}} )^{n−1−α} t_{0}^{j} (\frac{∂}{∂t_{1}} )^{j} p(t_{0}, t_{1}) and the constants c_{∗∗}^{∗} are given by

c_{j0}^{α} = \begin{cases}(α + 1)!(n + j − α − 1)!λ_{α+1−j}, for − 1 ≤ α − j ≤ n − 1 \\ 0, otherwise \end{cases}

and

c_{i1}^{ν} = \begin{cases}(α)!(n + i − α)!λ_{α−i}, for 0 ≤ α − i ≤ n \\ 0, otherwise \end{cases} .

The operator of multiplication by these forms

∂_{p} : \underbrace{S(1, 0) ⊕ ⋯ ⊕ S(1, 0)}_{n} → S(2, 1)

is represented by a square matrix of the size n(n − 1) and acts on the vectors of the above basis by the rule

∂_{p}(ψ_{k}^{(α)}) = ∑_{i=0}^{n−2} ψ_{k}ψ_{j}(c_{j0}^{α}t_{0} + c_{j1}^{α}t_{1}) = ∑_{j,l} ξ_{jkl}^{α}ψ_{j}ψ_{k}t_{l} ,

where

ξ_{jkl}^{α} = \begin{cases}(α + 1)!(n + j − α − 1)!λ_{α−j+1} for l = 0; −1 ≤ α − j ≤ n − 1 \\ (α)!(n + j − α)!λ_{α−j} for l = 1; 0 ≤ α − j ≤ n \\ 0, otherwise \end{cases}

and ν = 0, …, n − 1; j, k = 0, 1, … n − 2; l = 0, 1.

If we order ψ_{k}^{(α)} as

ψ_{0}^{0}, …, ψ_{n−1}^{0} , ψ_{0}^{1}, …, ψ_{n−2}^{1}, …, …, ψ_{0}^{n−1} , …, ψ_{n−2}^{n−1} ,

take ψ_{j}ψ_{k}t_{l} only with j ≤ k, and order them lexicographically, then, for example, for n = 3 we get

∂_{p} = (\begin{matrix}2!λ_{1} & 3!λ_{0} & 0 & 3!λ_{0} & 0 & 0 \\ 0 & 2λ_{1} & 3!λ_{0} & 0 & 3!λ_{0} & 0 \\ 2λ_{2} & 4λ_{1} & 0 & 2λ_{1} & 3!λ_{0} & 0 \\ 0 & 2λ_{2} & 4λ_{1} & 0 & 2λ_{1} & 3!λ_{0} \\ 3!λ_{3} & 3!λ_{2} & 0 & 2λ_{2} & 4λ_{1} & 0 \\ 0 & 3!λ_{3} & 3!λ_{2} & 0 & 2λ_{2} & 4λ_{1} \end{matrix}) ,

det ∂_{p} = 512(72λ_{1}^{4}λ_{0}λ_{2} − 189λ_{0}^{2}λ_{2}^{2}λ_{1}^{2} − 108λ_{0}^{2}λ_{1}^{3}λ_{3} − 729λ_{0}^{4}λ_{3}^{2} + 108λ_{0}^{3}λ_{2}^{3} − 8λ_{1}^{6} + 486λ_{0}^{3}λ_{2}λ_{1}λ_{3}) . For n = 4 the operator ∂_{p} has the matrix

∂_{p} = (\begin{matrix}6λ_{1} & 24λ_{0} & 0 & 0 & 0 & 0 & 24λ_{0} & 0 & 0 & 0 & 0 & 0 \\ 0 & 6λ_{1} & 0 & 24λ_{0} & 0 & 0 & 0 & 24λ_{0} & 0 & 0 & 0 & 0 \\ 0 & 0 & 6λ_{1} & 0 & 24λ_{0} & 0 & 0 & 0 & 24λ_{0} & 0 & 0 & 0 \\ 4λ_{2} & 12λ_{1} & 48λ_{0} & 0 & 0 & 0 & 6λ_{1} & 24λ_{0} & 0 & 0 & 0 & 0 \\ 0 & 4λ_{2} & 0 & 12λ_{1} & 48λ_{0} & 0 & 0 & 6λ_{1} & 0 & 24λ_{0} & 0 & 0 \\ 0 & 0 & 4λ_{2} & 0 & 12λ_{1} & 24λ_{0} & 0 & 0 & 6λ_{1} & 0 & 24λ_{0} & 0 \\ 6λ_{3} & 12λ_{2} & 36λ_{1} & 0 & 0 & 0 & 4λ_{2} & 12λ_{1} & 48λ_{0} & 0 & 0 & 0 \\ 0 & 6λ_{3} & 0 & 12λ_{2} & 36λ_{1} & 0 & 0 & 4λ_{2} & 0 & 12λ_{1} & 48λ_{0} & 0 \\ 0 & 0 & 6λ_{3} & 0 & 12λ_{2} & 36λ_{1} & 0 & 0 & 4λ_{2} & 0 & 12λ_{1} & 48λ_{0} \\ 24λ_{4} & 24λ_{3} & 48λ_{2} & 0 & 0 & 0 & 6λ_{3} & 12λ_{2} & 36λ_{1} & 0 & 0 & 0 \\ 0 & 24λ_{4} & 0 & 24λ_{3} & 48λ_{2} & 0 & 0 & 6λ_{3} & 0 & 12λ_{2} & 36λ_{1} & 0 \\ 0 & 0 & 24λ_{4} & 0 & 24λ_{3} & 48λ_{2} & 0 & 0 & 6λ_{3} & 0 & 12λ_{2} & 36λ_{1} \end{matrix})

and zero set of its determinant consists of all polynomials

p = λ_{0}t_{1}^{4} + λ_{1}t_{1}^{3}t_{0} + λ_{2}t_{1}^{2}t_{0}^{2} + λ_{3}t_{1}t_{0}^{3} + λ_{4}t_{0}^{4}

such that there exists a nontrivial linear combination α_{0}p + α_{1}p′ + α_{2}p′′, which has a zero of multiplicity 4.

2.3.3. Example. In the simplest nonextremal case Σ_{1} ⊂ P_{4} we get the following explicit expression for ∂_{p} (where the coordinates of ∂_{p}(ψ_{β}^{(α)}t_{0}^{γ}t_{1}^{1−γ}) will be written in rows to save space):

∂_{p} = (\begin{matrix}12λ_{0} & 6λ_{1} & 2λ_{2} & 0 & 0 & 24λ_{0} & 6λ_{1} & 0 & 0 & 0 & 0 & 0 \\ 0 & 12λ_{0} & 6λ_{1} & 2λ_{2} & 0 & 0 & 24λ_{0} & 6λ_{1} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 12λ_{0} & 6λ_{1} & 2λ_{2} & 0 & 0 & 24λ_{0} & 6λ_{1} & 0 \\ 0 & 0 & 0 & 0 & 0 & 12λ_{0} & 6λ_{1} & 2λ_{2} & 0 & 0 & 24λ_{0} & 6λ_{1} \\ 3λ_{1} & 4λ_{2} & 3λ_{3} & 0 & 12λ_{0} & 12λ_{1} & 6λ_{2} & 0 & 0 & 0 & 0 & 0 \\ 0 & 3λ_{1} & 4λ_{2} & 3λ_{3} & 0 & 12λ_{0} & 12λ_{1} & 6λ_{2} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 3λ_{1} & 4λ_{2} & 3λ_{3} & 0 & 12λ_{0} & 12λ_{1} & 6λ_{2} & 0 \\ 0 & 0 & 0 & 0 & 0 & 3λ_{1} & 4λ_{2} & 3λ_{3} & 0 & 12λ_{0} & 12λ_{1} & 6λ_{2} \\ 2λ_{2} & 6λ_{3} & 12λ_{4} & 0 & 6λ_{1} & 12λ_{2} & 12λ_{3} & 0 & 0 & 0 & 0 & 0 \\ 0 & 2λ_{2} & 6λ_{3} & 12λ_{4} & 0 & 6λ_{1} & 12λ_{2} & 12λ_{3} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 2λ_{2} & 6λ_{3} & 12λ_{4} & 0 & 6λ_{1} & 12λ_{2} & 12λ_{3} & 0 \\ 0 & 0 & 0 & 0 & 0 & 2λ_{2} & 6λ_{3} & 12λ_{4} & 0 & 6λ_{1} & 12λ_{2} & 12λ_{3} \end{matrix})

The zero set of the determinant det ∂_{p} consists of all polynomials

p = λ_{0}t_{1}^{4} + λ_{1}t_{1}^{3}t_{0} + λ_{2}t_{1}^{2}t_{0}^{2} + λ_{3}t_{1}t_{0}^{3} + λ_{4}t_{0}^{4}

such that there exists a nontrivial linear combination α_{0}p + α_{1}p′, which has a zero of multiplicity ≥ 3.

3. Affine geometric and topological properties of Σ_{i}

3.1. Differential operators. Our studying of the affine geometry of the hypersurface Σ_{i} ⊂ P_{n} is based on the (non-canonical) affine isomorphism between the spaces of monic polynomials and invertible linear differential operators with constant coefficients.

3.1.1. Notation. Consider the vector space V_{n} ⊂ ℂ[x] of all polynomials of degree ≤ n and denote by A_{n} ⊂ End_{ℂ}(V_{n}) the subalgebra of all linear endomorphisms commuting with all translations p(x) ↦ p(x − a). It is well known (see [3]) that A_{n} coincides with the truncated polynomial ring ℂ[D]/D^{n+1}, where D = d/d x. As in the previous section, we denote by D_{i} ⊂ A_{n} the subspace formed by differential operators of order less or equal i and by ℙ(D_{i}) the projectivization of D_{i}.

Let U_{n} ⊂ A_{n} be the multiplicative subgroup of all unipotent operators, i.e.

U_{n} = {1 + λ_{1}D + ⋯ + λ_{n}D^{n} mod(D^{n+1}) | λ_{k} ∈ ℂ} .

Note that all unipotent operators are invertible. For a subset M ⊂ U_{n} we will denote by M^{−1} = { m^{−1} | m ∈ M } the set of all inverse operators.

3.1.2. Affine identification of P_{n} with U_{n}. Let us define the affine isomorphism

ι : U_{n} ⟶ P_{n}

by taking a polynomial p ∈ P_{n} to the unique differential operator Ψ_{p} ∈ U_{n} such that p = Ψ_{p}(x^{n}). In the rest of the paper we will usually identify both spaces by this isomorphism.

Almost all our geometric constructions will be done in the space U_{n}. First of all, in the proposition 3.2. below we describe the i-th associated discriminant Σ_{i} in terms of U_{n}. In what follows we will denote this hypersurface in U_{n} by Σ_{i} as well and will freely use both versions Σ_{i} ⊂ P_{n} and Σ_{i} ⊂ U_{n} without special indication.

3.2. Proposition. The above isomorphism ι between U_{n} and P_{n} identifies Σ_{i} with the hypersurface in U_{n} consisting of all operators Ψ for which there exists a triple

(Ξ, Θ, a) ∈ ℙ(D_{i}) × ℙ(D_{n−i−2}) × ℂ

such that ΞΨ = Θ exp (−aD).

Proof. By definition, Σ_{i} consists of polynomials p ∈ P_{n} for which there exists a nontrivial i-tuple α_{0}, α_{1}, …, α_{i} such that α_{0}p + α_{1}p′ + ⋯ + α_{i}p^{(i)} has a zero of multiplicity at least i + 2 at some point a ∈ ℂ, i. e. α_{0}p + α_{1}p′ + ⋯ + α_{i}p^{(i)} = (x − a)^{i+2}q(x), where deg(q) ≤ n − i − 2. In terms of differential operators this means that there exists a pair of operators Ξ ∈ ℙ(D_{i}), Θ ∈ ℙ(D_{n−i−2}) and a point a ∈ ℂ such that Ξ(p) = Θ(x − a)^{n}.

Furthermore, the 1-parameter family (x − a)^{n} can be presented as the orbit of the polynomial x^{n} under the action of the 1-parameter subgroup exp (−aD) ⊂ U_{n}, where exp (aD) = ∑(aD)^{j}/j! ∈ U_{n}. Therefore the condition p ∈ Σ_{i} is equivalent to the existence of a triple

(Ξ, Θ, a) ∈ ℙ(D_{i}) × ℙ(D_{n−i−2}) × ℂ

such that p satisfies Ξ(p) = Θ exp (−aD)(x^{n}). The isomorphism ι : U_{n} → P_{n} identifies p with the unique Ψ_{p} such that p = Ψ_{p}(x^{n}). Thus p ∈ Σ_{i} if and only if ΞΨ_{p} = Θ exp (−aD). □

3.2.1. Corollary. The involution inv : Ψ → Ψ^{−1} of U_{n} induces the biregular isomorphism between the affine algebraic hypersurfaces Σ_{i} and Σ_{n−2−i} for all i = 0, …, n − 2.

Proof. We have to show that the involution inv : U_{n} → U_{n} which sends each operator Ψ ∈ U_{n} onto its inverse Ψ^{−1} maps Σ_{i} isomorphically onto Σ_{n−i−2}. Indeed, Ψ belongs to Σ_{i} if and only if there exists at least one triple

Ξ ∈ ℙ(D_{i}) , Θ ∈ ℙ(D_{n−i−2}) , a ∈ ℂ such that ΞΨ = Θ exp (−aD) .

The last identity is equivalent to Ξ exp (aD) = ΘΨ^{−1}, which exactly means that Ψ^{−1} belongs to Σ_{n−i−2}. □

3.2.2. Remark. The involution inv acts on the 1-parameter subgroup exp (aD) ⊂ U_{n} by changing the sign of a. So, under the above isomorphism between Σ_{i} and Σ_{n−2−i} polynomials with a multiple generalized zero at a point a ∈ ℂ are transformed into polynomials with a multiple generalized zero at the opposite point −a.

3.2.3. Remark. Since the projective hypersurfaces Σ̂_{i} and Σ̂_{n−i−2} (considered in the previous paragraph) have different degrees, the isomorphisms between Σ_{i} and Σ_{n−i−2} can not be extended to the regular isomorphisms of their projective closures and give only a birational equivalence.

3.3. Subdivision of Σ_{i} by Σ_{i,j}. One can easily extend the above results to more general loci Σ_{i,j} ⊂ P_{n}.

By the definition, Σ_{i,j} ⊂ P_{n} is the set of all polynomials p ∈ P_{n} such that some nontrivial linear combination α_{0}p + α_{1}p′ + ⋯ + α_{i}p^{(i)} has a zero of multiplicity greater or equal j + 2. Fixing a ∈ ℂ one gets Σ_{i,j}(a) ⊂ Σ_{i,j} consisting of all p ∈ Σ_{i,j} with a multiple i-generalized zero at a. We will call Σ_{i,j}(a) the fiber of Σ_{i,j} at the point a ∈ ℂ.

The locus Σ_{i,i} coincides with our original discriminant hypersurface Σ_{i}. In the general case, the codimension of Σ_{i,j} and Σ_{i,j}(a) in P_{n} is equal to j − i + 1 and j − i + 2 respectively. We have a natural stratification

Σ_{i} = Σ_{i,i} ⊃ Σ_{i,i+1} ⊃ ⋯ ⊃ Σ_{i,n−3} ⊃ Σ_{i,n−2} ⊃ ∅.

Obviously, Σ_{ij} ⊂ ⋂_{ν=i}^{j} Σ_{ν} = Σ_{i} ∩ Σ_{j}.

Exactly the same way as in Proposition 3.2. we get

3.3.1. Proposition. The stratum Σ_{i,j} is isomorphic to the set of all operators Ψ for which there exists a triple

(Ξ, Θ, a) ∈ ℙ(D_{i}) × ℙ(D_{n−j−2}) × ℂ

such that ΞΨ = Θ exp (−aD). □

3.3.2. Corollary. The involution inv : Ψ → Ψ^{−1} of the affine space U_{n} induces the biregular algebraic isomorphisms between Σ_{i,j} and Σ_{n−2−j,n−2−i} for all 0 ≤ i ≤ j ≤ n − 2 and between Σ_{i,j}(a) and Σ_{n−j−2,n−i−2}(−a) for all i + j + 1 ≥ n. □

3.4. Desingularization of Σ_{i,j}. Now we describe a natural resolution of singularities of Σ_{i,j}, which follows from prop. 3.3.1. Let

H_{i,j} ⊂ ℙ(D_{i}) × U_{n} × ℂ

be the subvariety of triples (Ξ, Ψ, a) such that ΞΨ = Θ exp (−aD) for at least one Θ ∈ ℙ(D_{n−i−2}).

We denote by H_{i,j}(a) the fiber of H_{i,j} over the point −a ∈ ℂ (i. e. the set of all the triples (Ξ, Ψ, −a) with the same a). Note that any H_{i,j}(a) can be naturally algebraically identified with H_{i,j}(0):

(Ξ, Ψ, −a) ∈ H_{i,j}(a) ⟺ (Ξ, Ψ exp(aD), 0) ∈ H_{i,j}(0) .

Hence, H_{i,j} = ℂ × H_{i,j}(0).

We fix the notation

π : H_{i,j} → Σ_{ij} ∈ U_{n}

for the natural projection of H_{i,j} onto the second factor in ℙ(D_{i}) × U_{n} × ℂ.

3.4.1. Proposition. H_{i,j}(0) is a smooth complete intersection of quadrics and thus the restriction

π|_{H_{i,j}(0)} : H_{i,j}(0) → Σ_{ij}(0)

gives a resolution of singularities of Σ_{i,j}(a). Moreover, this resolution is semismall, i. e. it satisfies the condition:

codim {p ∈ Σ_{i,j}(a) : dim π_{i}^{−1}(p) ≥ l} = 2l for all l .

Proof. A pair (Ξ, Ψ) = ( α_{0} + α_{1}D + ... + α_{i}D , 1 + ψ_{1}D + ... + ψ_{n}D^{n} ) presents a point

(Ξ, Ψ) ∈ H_{i,j}(0) if and only if the coefficients of degrees n − j − 1, ..., n in the expansion of

(α_{0} + α_{1}D + ... + α_{i}D)(1 + ψ_{1}D + ... + ψ_{n}D^{n}) ∈ A_{n}

vanish. Thus H_{i,j}(0) is defined in ℙ(D_{i}) × U_{n} by the system of j + 2 quadratic equations in variables (α_{0}, …, α_{i}, ψ_{1}, … ψ_{n}). In the case i + j + 1 < n this system has the form

{α_{0}ψ_{n} + α_{1}ψ_{n−1} + ⋯ + α_{i}ψ_{n−i} = 0 ; α_{0}ψ_{n−1} + α_{1}ψ_{n−2} + ⋯ + α_{i}ψ_{n−i−1} = 0 ; ⋮ ; α_{0}ψ_{n−j−1} + α_{1}ψ_{n−j−2} + ⋯ + α_{i}ψ_{n−i−j−1} = 0 .

The Jacobi matrix J of this system is the (j + 2) × (2i + j + 3) matrix of the form

(\begin{matrix}ψ_{n} & ψ_{n−1} & … & ψ_{n−i} & 0 & … & 0 & α_{i} & α_{i−1} & … & α_{0} \\ ψ_{n−1} & ψ_{n−2} & … & ψ_{n−i−1} & ⋮ & ... & α_{i} & α_{i−1} & … & α_{0} & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & 0 & ... & ... & ... & ... & ... & ⋮ \\ ψ_{n−j−1} & … & … & ψ_{n−i−j−1} & α_{i} & α_{i−1} & … & α_{0} & 0 & … & 0 \end{matrix} ) .

Since (α_{0}, α_{1}, …, α_{i}) ≠ (0, 0, …, 0) the rank of J is always equal to j + 2. This means that H_{i,j}(0) is a smooth complete intersection. The case i + j + 1 ≥ n is analogous. The Jacobi matrix contains a unit submatrix and a complementary submatrix of maximal rank.

Let us now consider the map π : H_{i,j}(0) → Σ_{i,j}(0). The inverse image π^{−1}(p) of any polynomial p ∈ Σ_{i,j}(0) is a projective subspace. One has that dim(π^{−1}(p)) equals to the corank of the above linear system reduced by 1. One can easily show that for a generic Ψ ∈ Σ_{i,j}(0) the corank of the system equals 1 and one gets that H_{i,j}(0) is a resolution of singularities of Σ_{i,j}(0). Moreover, more detailed consideration of the linear system (or of the space of Hankel matrices, see Section 4) shows that the set {p ∈ Σ_{i,j}(a) : dim π^{−1}(p) ≥ l} has codimension 2l in Σ_{i,j}(a) and a. □

3.4.2. Corollary. π : H_{i,j} → Σ_{ij} is a resolution of singularities of Σ_{i,j}.

3.4.3. Remark. In the case i+j +1 < n the set H_{i,j}(0) has a nonsingular projective closure

Ĥ_{i,j}(0) ⊂ ℙ(D_{i}) × ℙV_{n} .

Ĥ_{i,j}(0) also is a smooth complete intersection of quadrics and one can calculate its cohomology as well as that of H_{i,j}(0).

The existence of a semismall resolution also gives a hope that one can calculate the intersection homology of Σ_{i,j}(0). In the case i + j + 1 ≥ n the variety Ĥ_{i,j}(0) is singular at infinity with a conic type of singularities.

3.5. Topology of the fiber Σ_{i,j}(0). Here we are going to clarify the topology of the zero fiber Σ_{i,j}(0). Recall that in terms of the space U_{n} of unipotent differential operators the space Σ_{i,j}(0) consists of all operators Ψ ∈ U_{n} such that

ΞΨ = Θ for some Ξ ∈ ℙ(D_{i}) , Θ ∈ ℙ(D_{n−j−2}).

Consider the decreasing filtration

Σ_{i,j}(0) = Σ_{i,j}^{0}(0) ⊃ Σ_{i,j}^{1}(0) ⊃ ⋯ ⊃ Σ_{i,j}^{i}(0) ,

where Σ_{i,j}^{l}(0) consists of all Ψ ∈ Σ_{i,j}(0) such that any Ξ ∈ D_{i} \ 0 with the property that ΞΨ ∈ D_{n−j−2} is divisible by D^{l}.

In terms of the Proposition 3.4.1, the closed subset Σ_{i,j}^{l}(0) ⊂ Σ_{i,j}(0) consists of all Ψ such that the projection of the fiber π^{−1}(Ψ) ⊂ H_{i,j} ⊂ ℙ(D_{i}) × U_{n} onto the first factor ℙ(D_{i}) is contained in the subspace ℙ(D^{l} · D_{i}) ⊂ ℙ(D_{i}).

3.5.1. Lemma. The difference Σ_{i,j}^{0}(0) \ Σ_{i,j}^{1}(0) is isomorphic to the space CoPr(i, n − i − 2) of all pairs of coprime polynomials

( 1 + α_{1}D + ⋯ + α_{r}D^{r} , 1 + ν_{1}D + ⋯ + ν_{s}D^{s} )

such that r ≤ i, s ≤ (n − j − 2).

Proof. We are going to verify that the complement

Σ_{i,j}^{0}(0) \ Σ_{i,j}^{1}(0) ⊂ Σ_{i,j}(0)

consists of all Ψ(D) = A^{−1}(D)B(D), where A ∈ U_{n} ∩ D_{i}, B ∈ U_{n} ∩ D_{n−j−2} are any two coprime (in ℂ[D]) polynomials, and that Ψ and (A, B) are uniquely determined by each other.

If Ψ ∈ Σ_{i,j}^{0}(0) \ Σ_{i,j}^{1}(0) then there exists at least one pair

A′ = 1 + α_{1}D + ⋯ + α_{i}D^{i} , B′ = 1 + ν_{1}D + ⋯ + ν_{n−j−2}D^{n−j−2}

such that ΨA′ = B′. If A′ and B′ are not coprime then A′ = HA, B′ = HB, where H = 1 + h_{1}D + ⋯ + h_{m}D^{m} is their maximal common factor. Since H is invertible in U_{n}, one gets ΨA = B and Ψ = A^{−1}B is represented in the required form.

On the other side, if A ∈ U_{n} ∩ D_{i} and B ∈ U_{n} ∩ D_{n−j−2} are coprime then the quotient Ψ = A^{−1}B, obviously, belongs to Σ_{i,j}^{0}(0)Σ_{i,j}^{1}(0) and it remains to show that this quotient in U_{n} determines A ∈ U_{n} ∩ D_{i} and B ∈ U_{n} ∩ D_{n−j−2} uniquely. Let us assume that for some A_{1} ∈ U_{n} ∩ D_{i}, B_{1} ∈ U_{n} ∩ D_{n−j−2}. We have A_{1}^{−1}B_{1} = A^{−1}B in U_{n} and after multiplying by A_{1}A in U_{n} we get the identity A_{1}B = AB_{1}, which holds in the usual polynomial ring ℂ[D]. Since A and B are coprime, therefore A_{1} is divisible by A. Thus A_{1} =A and hence B_{1} =B. □

3.5.2. Lemma. The homomorphism of reduction modulo D^{n−k+1}:

res_{n,k} : U_{n} → U_{n−k} ,

which sends A(D) mod (D^{n+1}) ↦ A(D) mod (D^{n−k+1}), maps the k-th complement

Σ_{i,j}^{k}(0) \ Σ_{i,j}^{k+1}(0)  in  Σ_{i,j}(0) ⊂ U_{n}

epimorphically onto the 0-th complement

Σ_{i,j}^{0}(0) \ Σ_{i,j}^{1}(0)  in  Σ_{i−k,j}(0) ⊂ U_{n−k}

and the corresponding restriction

Σ_{i,j}^{k}(0) \ Σ_{i,j}^{k+1}(0) ⟶^{res_{n,k}} Σ_{i,j}^{0}(0) \ Σ_{i,j}^{1}(0)

is the trivial bundle with the fiber ℂ^{k}.

Proof. If Ψ ∈ Σ_{i,j}^{k}(0) \ Σ_{i,j}^{k+1}(0) ⊂ Σ_{i,j}(0) ⊂ U_{n}, then

D^{k}AΨ = BD^{k} in U_{n}

for some A ∈ U_{n} ∩ D_{i−k}, B ∈ U_{n} ∩ D_{n−j−2}. Hence, in U_{n−k} we have AΨ̃ = B, where Ψ̃ = res_{n,k}Ψ and

res_{n,k}(Σ_{i,j}^{k}(0) \ Σ_{i,j}^{k+1}(0)) ⊂ Σ_{i,j}^{0}(0) \ Σ_{i,j}^{1}(0).

To finish the proof, we describe the complete pullback res_{n,k}^{−1}(Σ_{i,j}^{0}(0) \ Σ_{i,j}^{1}(0)). Let Ψ̃ ∈

Σ_{i,j}^{0}(0) \ Σ_{i,j}^{1}(0) inside Σ_{i−k,j}(0) ⊂ U_{n−k}. As we have seen above, AΨ̃ = B in U_{n−k} for some A ∈ U_{n−k} ∩ D_{i−k}, B ∈ U_{n−k} ∩ D_{n−j−2}. If we consider these A, B as elements of U_{n}, then for any Ψ ∈ U_{n} such that res_{n,k}Ψ = Ψ̃ we get D^{k}ΨA = D^{k}B in U_{n}. Thus, Ψ belongs to Σ_{i,j}^{k}(0) ∈ U_{n} and

res_{n,k}^{−1}(Σ_{i,j}^{0}(0) \ Σ_{i,j}^{1}(0)) ⊂ Σ_{i,j}^{k}(0) ⊂ Σ_{i} ⊂ U_{n}.

It remains to prove that the left side locus lies inside Σ_{i,j}^{k}(0) \ Σ_{i,j}^{k+1}(0). But if above Ψ has the property Ψ ∈ Σ_{i,j}^{k+1}(0) then we get the relation

D^{k+1}ΨA_{1} = D^{k+1}B in U_{n}.

After the reduction modulo D^{n−k+1} it takes the form Ψ̃DA_{1} = DB_{1}, which contradicts to the condition Ψ̃ ∉ Σ_{i,j}^{1}(0). □

3.5.3. Topology of CoP r(r, s). The space CoPr(r, s) can be also interpreted as the space of all pairs of monic polynomials of degree r and s with no common roots possibly except for the origin. (The possible multiplicity of a common zero at the origin can vary from 0 to min(r, s).) Note that the cohomology of the space of all pairs of coprime monic polynomials of some given degrees is known, see e.g. [9] but is not of any immediate help for the space CoPr(r, s). It would be very interesting to calculate the cohomology of the space CoPr(r, s) and to compare it with that of the usual space of rational functions and the braid space, see e.g. [10]. But, unfortunately, the complete information about the cohomology of Σ_{i,j}(0) is unavailable at the present moment. This makes it impossible to calculate the cohomology even for the case of stable strata which have the simplest form among Σ_{i,j}, (see 3.7 below).

Conjecture. The Leray spectral sequence of the above filtrations with Z-coefficients collapses at the E_{2}-term.

3.6. Rationality of Σ_{i,j}. In this subsection we will prove the rationality of Σ_{i,j} using the filtration of the fiber Σ_{i,j}(0) considered in the previous section.

3.6.1. Lemma. Σ_{i,j}(0) is a rational variety.

Proof. In fact, a birational equivalence between Σ_{i,j}(0) and ℙ(D_{i}) × ℙ(D_{n−j−2}) is given in the proof of Lemma 3.5.1. Namely, Σ_{i,j}(0) contains a Zariski open subset of all pairs of coprime polynomials

( 1 + α_{1}D + ⋯ + α_{i}D^{i} , 1 + μ_{1}D + ⋯ + μ_{n−j−2}D^{n−j−2} ) ,

which have the degrees exactly equal to i and n − j − 2. □

The following proposition implies the rationality of Σ_{i,j}.

3.6.2. Proposition. There exists a Zariski open subset in the space ℙ(D_{i}) × ℙ(D_{n−j−2}) × ℂ isomorphic to a Zariski open subset in Σ_{i,j} and therefore Σ_{i,j} is rational.

Proof. As above we have a family of codimension 1 subvarieties Σ_{i,j}(a), a ∈ ℂ in Σ_{i,j}, such that for all a_{1} ≠ a_{2}, Σ_{i,j}(a_{1}) and Σ_{i,j}(a_{2}) are isomorphic. By Propositon 3.6.1 every Σ_{i,j}(a) is a rational variety. Since group of affine transformations x → cx + d acts bitransitively on ℂ, to prove rationality of Σ_{i,j} it suffices to show that codim_{ℂ} (Σ_{i,j}(0) ∩ Σ_{i,j}(1)) in Σ_{i,j}(0) is at least 2. This will imply that for a Zariski open subset in Σ_{i,j} there exists exactly one triple (Ξ, Θ, a) ∈ ℙ(D_{i}) × ℙ(D_{n−j−2}) × ℂ such that ΞΨ_{p} = Θe^{aD}.

Indeed, consider Ω = ⋃_{a_{2}∈ℂ\a_{1}}(Σ_{i,j}(a_{1}) ∩ Σ_{i,j}(a_{2})). If codim_{ℂ}(Σ_{i,j}(0)) ∩ Σ_{i,j}(1)) in Σ_{i,j}(0)

is at least 2 then Ω has codimension at least 1 in Σ_{i,j}(a) and its complement is a Zariski open subset. Varying a_{1} and taking the union of the complements we get a Zariski open subset in the whole Σ_{i,j} for which there exists only one triple (Ξ, Θ, a) such that ΞΨ_{p} = Θe^{aD}. The following lemma accomplishes the proof. □

3.6.3. Lemma. For any pair (a_{1}, a_{2}), (a_{1} ≠ a_{2}) the intersection Σ_{i,j}(a_{1}) ∩ Σ_{i,j}(a_{2}) has codimension 2 in Σ_{i,j}.

Proof. As was mentioned before codim(Σ_{i,j}(a_{1}) ∩ Σ_{i,j}(a_{2})) does not depend on particular choice of (a_{1}, a_{2}). Now let Σ̂_{i,j}(a) ⊂ P̂_{n} be the projective closure of Σ_{i,j}(a) ⊂ P_{n} and let Σ̂_{i,j} ⊂ P̂_{n} be the projective closure of Σ_{i,j}. One can easily see that the fiber at infinity Σ̂_{i,j}(∞) = Σ̂_{i,j} \ Σ_{i,j} is a projective subspace of dimension n − 2 − j + i.

Take the correspondence Cor in the space of Σ_{i,j} × ℂ consisting of all pairs (Ψ, α) where Ψ ∈ Σ_{i,j} and α is a generalized zero of Ψ and consider its closure Ĉor ⊂ Σ̂_{i,j} × ℙ_{1}. The inverse image of any point a ∈ ℙ_{1} is Σ̂_{i,j}(a). It suffices to show that codim(Σ̂_{i,j}(∞)∩Σ_{i,j}(a_{2})) in Σ_{i,j} equals 2 since the codimension of intersection of any two fibers gives the upper bound for the codimension of intersection of two generic fibers.

Notice that Σ̂_{i,j}(a) ∩ P̂^{l} ⊂ P̂^{n} is isomorphic to Σ̂_{i,j}(a) ⊂ P̂^{l} where P̂^{l} is the projectivized space of polynomials of degree at most l in t_{0} : t_{1}. Since the special fiber Σ̂_{i,j}(∞) is isomorphic to P̂^{n−j+i−2} one gets that Σ̂_{i,j}(∞) ∩ Σ̂_{i,j}(a_{2}) ⊂ P̂^{n} is isomorphic to Σ̂_{i,j}(a_{2}) ⊂ P̂^{n−j+i−2}. The latter space, obviously, has codimension 1 in Σ̂_{i,j}(∞). Thus the former space has codimension 2 in Σ̂_{i,j}. □

3.6.4. Lemma. For a generic k-tuple (a_{1}, …, a_{k}) (a_{1} ≠ a_{2} ≠ ⋯ ≠ a_{k}), k ≤ \frac{n}{j−i+2}, the intersection Σ_{i,j}(a_{1}) ∩ Σ_{i,j}(a_{2}) ∩ ⋯ ∩ Σ_{i,j}(a_{k}) has codimension k in Σ_{i,j}.

Proof. We are going to iterate the above arguments using induction on the number of points and dimension. Namely, assume that we have proved that the codimension is the expected one for a Zariski open set in the space of (k − 1)-tuples and for all m < n. For generic a_{1}, a_{2}, …, a_{k} one has

codim (Σ_{i,j}(a_{1}) ∩ Σ_{i,j}(a_{2}) ∩ ⋯ ∩ Σ_{i,j}(a_{k})) ≤ codim (Σ̂_{i,j}(∞) ∩ Σ_{i,j}(a_{2}) ∩ ⋯ ∩ Σ_{i,j}(a_{k})) .

But the later intersection is isomorphic to the intersection of (k − 1)-tuples in the space P̃^{n−j+i−2} and thus has the expected codimension by the inductive assumption. □

3.6.5. Remark. It is worth mentioning that not for all a_{1}, a_{2}, … a_{k}, k > 2, the intersection Σ_{i,j}(a_{1}) ∩ Σ_{i,j}(a_{2}) ∩ ⋯ ∩ Σ_{i,j}(a_{k}) has the expected codimension. Namely, there exist polynomials p ∈ P_{n} which have more than [n/2] i-generalized multiple zeros. This means that the intersection of more than the expected number of Σ_{i}(a_{l}) is still nonempty, see Section 1. It would be very interesting to describe such ’Weierstrass’ k-tuples.

3.7. Some special cases. A stratum Σ_{i,j} ⊂ P_{n} is called stable if j −i+2 > [\frac{n}{2}] and unstable otherwise. The following proposition reduces the study of the topology of any stable Σ_{i,j} to that of its Σ_{i,j}(0).

3.7.1. Proposition. Any stable stratum Σ_{i,j} ⊂ U_{n} is isomorphic to Σ_{i,j}(0) × ℂ.

Proof. One has to show that if j − i + 2 > [n/2] then each p ∈ Σ_{i,j} belongs to the unique Σ_{i,j}(a), i.e. that the intersection Σ_{i,j}(a_{1}) ∩ Σ_{i,j}(a_{2}), a_{1} ≠ a_{2} is empty. This can be proved exactly along the same lines as those of the lemma in 3.6.3. □

3.7.2. Remark. The stable stratum Σ̂_{0,[ \frac{n}{2} ]−1} ⊂ P_{n} is called the nullcone and is of a fundamental importance in the representation theory of SL_{2}(ℂ). The following problem was formulated to the authors by J.Weyman in April 1995.

Question. Is it true that Σ̂_{0,[ \frac{n}{2} ]−1} is a set-theoretical complete intersection?

One can even speculate that set-theoretically one has Σ̂_{0,[ \frac{n}{2} ]−1} = ⋂_{i+0}^{[ \frac{n}{2} ]−1} Σ̂_{i}.

4. Example: cohomology of Σ_{1} ⊂ P_{4}

4.1. Vassiliev’s resolution. A natural approach to the problem how to calculate the cohomology of Σ_{i} ⊂ P_{n} with compact supports is to try to generalize the simplicial resolution which was succesfully used by V.Vassiliev for the series of strata Σ_{0}, see [10] (recall that P_{n}\Σ_{0} is the usual braid space). Such a generalized Vassiliev’s resolving space for Σ_{i} coincides with the set of all pairs {p, S_{p}}, where p ∈ Σ_{i} and S_{p} is the formal simplex spanned by all pairwise different multiple i-generalized roots of p. There exists a natural geometric realization of Σ̄_{i} in ℂ^{N} for some sufficiently big N.

By construction, this resolving space Σ̄_{i} has a natural filtration

Σ̄_{i} = F_{m} ⊃ F_{m−1} ⊃ ⋯ ⊃ F_{1} ,

where F_{i} is the set of all pairs {p, S_{p}^{i}}, p ∈ Σ_{i}, S_{p}^{i} denotes the (i − 1)-skeleton of S_{p}, and m is the maximal possible number of multiple pairwise different i-generalized zeros. (We use the convention that if dim S_{p} ≤ i then S_{p}^{i} = S_{p}.) Note that F_{1} is homeomorphic to Σ_{i}(0) × ℂ.

One can easily see that the obvious projection Σ̄_{i} → Σ_{i} is a homotopy equivalence extendable to their 1-point compactifications. Therefore, H_{c}^{∗}(Σ_{i}) = H_{c}^{∗}(Σ̄_{i}).

We will apply this program to Σ_{1} ⊂ P_{4}. In this case we have only the 3-term filtration

Σ̄_{1} = F_{3} ⊃ F_{2} ⊃ F_{1} ,

since the maximal number of multiple 1-generalized roots of a quartic polynomial equals 3, see 1.3. The main result of this section is the following

4.2. Proposition. The cohomology with compact supports of Σ_{1} ⊂ P_{4} is equal

( 0, 0, 0, 0, 0, ℤ_{2} ⊕ ℤ_{2}, ℤ ) .

The proof of this proposition splits into a series of lemmas and will be finished in 4.6 below.

4.3. Σ_{i}(0) and Hankel matrices. Recall that a matrix is called Hankel if it has the same entry on each anti-diagonal. Let us denote by Hank(k, l) the set of all (k ×l) Hankel matrices with complex coefficients and we denote by Hank_{deg}(k, l) ⊂ Hank(k, l), k ≤ l the subset of matrices of rank < k.

4.3.1. Lemma. If i+j +1 < n then Σ_{i,j}(a) is isomorphic to Hank_{deg}(i+1, j +2)×ℂ^{n−i−j−2}. If i + j + 1 ≥ n then Σ_{i,j}(a) is isomorphic to Σ_{n−j−2,n−i−2}(−a).

Proof. Since Σ_{i,j}(a) is isomorphic to Σ_{i,j}(0) we consider only the last case. For Σ_{i,j}(0) the statement follows directly from the definition. □

Now we can describe the cohomology with compact supports of the first term of the filtration F_{1} = Σ_{1}(0) × ℂ.

4.3.2. Lemma. The cohomology with compact supports of the variety Hank_{deg}(2, n) is equal to (0, 0, ℤ_{n}, 0, ℤ).

Proof. The set Hank_{deg}(2, n) in question is the cone with the vertex at the origin 0 over the rational normal scroll in ℂ^{n+1}. The set Hank_{deg}(2, n) \ 0 has the structure of a ℂ^{∗}-bundle over this rational normal scroll ℂℙ^{1}. The first Chern class c_{1} of this bundle is equal to the degree of the scroll and therefore c_{1} = n. Thus the usual cohomology of Hank_{deg}(2, n) \ 0 is equal to (ℤ, 0, ℤ_{n}, ℤ). Since it is a manifold its cohomology with compact supports is dual to the usual cohomology and is equal (0, ℤ, ℤ_{n}, 0, ℤ). From the long exact sequence

H_{c}^{∗}(Hank_{deg}(2, n)) → H_{c}^{∗}(Hank_{deg}(2, n)) \ 0 → H_{c}^{∗}(0)

one gets that H_{c}^{∗}(Hank_{deg}(2, n)) = (0, 0, ℤ_{n}, 0, ℤ). □

4.3.3. Corollary. H_{c}^{∗}(F_{1}) = (0, 0, 0, 0, ℤ_{3}, 0, ℤ).

Proof. F_{1} is homeomorphic to Hank_{deg}(2, 3) × ℂ. □

4.4. Cohomology of F_{2} \ F_{1}. We are going to describe explicitly the set of all quartic polynomials, which have exactly two 1-generalized roots at given points a and b in ℂ. We use here the technique and notation from 3.4 and 3.5.

4.4.1. Lemma. For any pair of distinct points (a, b) there exist exactly 4 polynomials in Σ_{1} ⊂ P_{4} such that their 1-generalized multiple zeros are a and b.

Proof. As above, it suffices to consider the case a = 0, b = t. Denote by Σ_{1}^{open}(t) a Zariski open subset in Σ_{1}(t) consisting of all Ψ = A^{−1}(D)B(D) exp(tD), where A(D) = 1 + αD, B = 1 + βD, and α ≠ β. For t = 0 the set Σ_{1}^{open}(t) is nothing more than the complement Σ_{1}^{0}(0) \ Σ_{1}^{1}(0) considered in 3.5.

In the case Σ_{1} ⊂ A_{4} the intersection Σ_{1}(0)∩Σ_{1}(t) is, obviously, contained inside Σ_{1}^{open}(0)∩ Σ_{1}^{open}(t). Thus, Ψ = A_{1}^{−1}(D)B_{1}(D) belongs to Σ_{1}^{open}(0) ∩ Σ_{1}^{open}(t) if and only if for some unipotent linear differential operators A_{2}(D), B_{2}(D) one gets

A_{1}^{−1}(D)B_{1}(D) = e^{tD}A_{2}^{−1}(D)B_{2}(D)

or, equivalently, A_{2}(D)B_{1}(D) = e^{tD}A_{1}(D)B_{2}(D).

To determine A_{1}, A_{2}, B_{1}.B_{2} we have to look at the intersection D_{2} ∩ exp(tD)D_{2}. Direct computation shows that this intersection is a 1-dimensional subspace in D_{2} spanned by the polynomial

1 + tD/2 + t^{2}D^{2}/12 = (1 + ϱtD)(1 + ϱ̄tD) ,

where −ϱ, −ϱ̄ are the complex conjugate roots of the quadratic polynomial x^{2} + x/2 + 1/12, i. e. ϱ = (3 + i\sqrt{3})/12, and we have

(1 + ϱtD)(1 + ϱ̄tD) = exp(tD) · (1 − ϱtD)(1 − ϱ̄tD) .

Thus Σ_{1}(0) ∩ Σ_{1}(t) consists of the following four operators:

\begin{aligned}(1 + ϱtD)(1 − ϱtD)^{−1} & = 1 + \frac{3 + i\sqrt{3}}{6}tD + \frac{1 + i\sqrt{3}}{12}t^{2}D^{2} + \frac{i\sqrt{3}}{36}t^{3}D^{3} + \frac{−1 + i\sqrt{3}}{144}t^{4}D^{4} \\ (1 + ϱ̄tD)(1 − ϱ̄tD)^{−1} & = 1 + \frac{3 − i\sqrt{3}}{6}tD + \frac{1 − i\sqrt{3}}{12}t^{2}D^{2} − \frac{i\sqrt{3}}{36}t^{3}D^{3} + \frac{−1 − i\sqrt{3}}{144}t^{4}D^{4} \\ (1 + ϱtD)(1 − ϱ̄tD)^{−1} & = 1 + \frac{1}{2}tD + \frac{3 − i\sqrt{3}}{24}t^{2}D^{2} + \frac{1 − i\sqrt{3}}{48}t^{3}D^{3} − \frac{i\sqrt{3}}{144}t^{4}D^{4} \\ (1 + ϱ̄tD)(1 − ϱtD)^{−1} & = 1 + \frac{1}{2}tD + \frac{3 + i\sqrt{3}}{24}t^{2}D^{2} + \frac{1 + i\sqrt{3}}{48}t^{3}D^{3} − \frac{i\sqrt{3}}{144}t^{4}D^{4} \end{aligned}

□

The previous formulas give also the exact description of how the fundamental group of the space of all pairs (a, b), a ≠ b, acts on the constructed quadruples of the polynomials. Namely, the interchange of a and b via the standard ℤ-generator of this fundamental group leads, obviously, to the inversion of the above four differential operators and change of the sign of t. This procedure preserves the first and the second of the above operators and interchanges the last two of them. We get

4.4.2. Corollary. The four polynomials from Lemma 4.4.1. are naturally organized in two pairs such that the replacement of (a, b) by (b, a) (via the standard generator of the fundamental group of the space of all pairs (a, b) such that a ≠ b) preserves both polynomials in one pair and interchanges polynomials in the other pair.

Now we can easily calculate the cohomology of F_{2} \ F_{1}.

4.4.3. Lemma. The cohomology with compact supports of F_{2} \ F_{1} is

(0, 0, 0, 0, ℤ, ℤ ⊕ ℤ_{2} ⊕ ℤ_{2}) .

Proof. Indeed, the set F_{2} \ F_{1} is a bundle over the set of all unordered distinct pairs of points with the fiber consisting of 4 segments. The action of the generator of the fundamental group of the base inverts the orientation of two of these segments and interchanges the two others with the inversion of orientation. Thus our fiber bundle consists of two copies of a 3-dimensional cylinder over a Möbius band and a 3-dimensional cylinder over an annulus.

Cohomology with compact supports of a Möbius band and an annulus is equal to (0, 0, ℤ_{2}) and (0, ℤ, ℤ) resp. Thus, H_{c}^{∗}(F_{2} \ F_{1}) is equal to

(0, 0, 0, 0, ℤ, ℤ ⊕ ℤ_{2} ⊕ ℤ_{2}) .  □

4.5. Cohomology of F_{3} \ F_{2} is also calculated via the exact description of all polynomials having three 1-generalized roots.

Lemma. The intersection Σ_{1}(a)∩Σ_{1}(b)∩Σ_{1}(c) for a triple of pairwise different points (a, b, c) is nonempty if and only if there exist two constants d and e ≠ 0 such that a − d, b − d, c − d are 3 different roots of the equation x^{3} = e. In this case Σ_{1}(a) ∩ Σ_{1}(b) ∩ Σ_{1}(c) contains exactly one polynomial (x + d)^{4} − 2ex.

Proof. By Proposition 1.3.1 a polynomial p ∈ Σ_{1} ⊂ P_{4} has at most three 1-generalized multiple zeros. Thus p belongs to at most some triple intersection Σ_{1}(a)∩Σ_{1}(b)∩Σ_{1}(c). Now let us determine the set of all p which have exactly 3 pairwise different multiple 1-generalized zeros. This is equivalent to finding such p that Δ_{1}(p) has 3 different double zeros which are not multiple zeros of p itself. Using the action on the affine group we can assume that p = x^{4} + λ_{1}x^{2} + λ_{2}x + λ_{3} which gives

Δ_{1}(p) = −4 [x^{6} + \frac{λ_{1}}{2} x^{4} − λ_{2}x^{3} + (\frac{λ_{1}^{2}}{2} − 3λ_{3}) x^{2} + \frac{λ_{1}λ_{2}}{2} x − \frac{λ_{1}λ_{3}}{2} + \frac{λ_{2}^{2}}{4} ] .

Δ_{1}(p) has 3 zeros if and only if Δ_{1}(p) = [(x+α)(x+β)(x−α−β)]^{2} for some α ≠ β ≠ −α−β. (Omitted) consideration of the last condition implies that this is possible if and only if p = x^{4} + λ_{2}x. In this case Δ_{1}(p) = −4(x^{3} − \frac{λ_{2}}{2} )^{2} with 3 pairwise different double zeros. This implies that Σ_{1}(a) ∩ Σ_{1}(b) ∩ Σ_{1}(c) is nonempty (and consists of exactly 1 polynomial (x + d)^{4} − 2ϵ) iff for some d the numbers a − d, b − d, c − d are roots of x^{3} = ϵ, ϵ ≠ 0. □

4.5.1. Corollary. The cohomology with compact supports of F_{3} \ F_{2} is equal

(0, 0, 0, 0, 0, ℤ, ℤ) .

Proof. Indeed, F_{3} \ F_{2} is a bundle over the set of all equilateral triangles on ℂ with the fiber consisting of the simplex formally spanned by triangle’s vertices. The base is homeomorphic to ℂ × ℝ^{+} × S^{1}. The generator of S^{1} of the base acts trivially on the orientation of the fiber since it induces the cyclic shift of vertices. Thus F_{3} \ F_{2} is homeomorphic to ℝ^{5} × S^{1}. □

4.6. End of the proof of Proposition 4.2. We accomplish the proof of Proposition 4.2 by using the Leray spectral sequence for the cohomology with compact supports for the whole Σ̄_{1}. Its E_{1}- and E_{2}- pages are given on Fig. 1. (Potentially nontrivial differentials are shown by arrows.)

4.6.1. Lemma. a) The differential ℤ → ℤ in the 3rd row of E_{1} is multiplication by 3;

b) the differential ℤ → ℤ in the 4th row of E_{1} is an isomorphism,

c) the differential ℤ_{3} → ℤ_{3} on E_{2} is an isomorphism.

Proof. We consider the boundary map for the corresponding homology cycles. By Lemma 4.4 the part F̃_{2} of F_{2} \F_{1} to which F_{3} \F_{2} is glued is a bundle over the braid space Br(2) with the fiber consisting of 2 segments which can be interpreted as formal simplices spanned by the ordered pairs of points. Under the action of π_{1} they change places with the inversion of their orientations. The 4-dimensional relative homology cycle Θ_{1} pairing with the 4-dimensional cohomology class in F_{2} \ F_{1} is obtained by taking fibers over all horizontal pairs of points in Br(2). Analogously, the 5-dimensional relative homology class Θ_{2} in F_{3} \ F_{2} pairing with the 5-dimensional generator in cohomology consists of all formal 2-simplices over all equilateral triangles with one horizontal side. One easily gets ∂(Θ_{2}) = 3Θ_{1} and a) follows. At the same time the boundary of the fundamental cycle in F_{3}\F_{2} equals F̃_{2} and b) follows. More detailed consideration of the boundary maps implies c). □

Figure 1: The first two pages in the calculation of cohomology of Σ_{1}

5. Final remarks

The above partial results on the algebraic and topological properties of the associated discriminants show that their structure is substantially more complicated than that of the classical discriminants.

Below we formulate a few (of many more) questions which seem natural in the context of the associated discriminants.

1) Calculate the number of connected components to the union of the associated discriminants in the space P_{n} with real coefficients and, more generally, for the space of solutions to some linear homogeneous ordinary differential equation with constant coefficients. (This question is the most interesting from the point of view of its application to Toda lattices.)

2) Calculate the usual and intersection (co)homology of Σ_{i}(0) or equivalently of the space Hank_{deg}(i, i). (For the usual cohomology one can use the stratification described in 3.5. For the intersection homology one can use the semismall desingularization. Many analogous spaces of matrices were sucessfully studied before. Recently Prof. R. MacPherson has suggested to the second author a recursive procedure applicable to determination of intersection homology of the space of degenerate Hankel matrices.)

3) Calculate the maximal number of pairwise different multiple i-generalized zeros for polynomials of degree n.

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Received September 4, 1996