- Email: [email protected]
Topology of complements of strata of the discriminant of polynomials
C. R. Acad. Sci. Paris, t. 327, Topologie/Topology (CComCtrie alghbrique/Algebraic
SCrie
I, p. 665-670,
1998
Geometry)
Topology of complements of strata of the discriminant of polynomials Fabien , Ecole
NAPOLITANO normale
E-mail: (Repu
suptrieure,
45, rue d’Ulm,
75005 Paris,
France
[email protected] le
9 juin
Abstract.
1998,
accept6
le
18
septembre
1998)
The discriminantof the set of unitary complex polynomialsof degreen is the variety of polynomialshaving at leastone doubleroot. For n Xarge enough,the set of rootsof a polynomial may have more complicatedsingularities.The discriminantis stratified by the varietiesof polynomialshaving given singularities.We study the cohomology groups of the complementof any stratumand we constructa cellular decomposition of the symmetric n-th power of C “compatible” with the stratification. Using this decomposition,one can computeby hand the first cohomologygroups.0 AcadCmie des Sciences/Elsevier,Paris Topologie du complbment des polyn6mes
R&urn&
des strates da. discriminant
Le discriminant de l’ensemble des polyn6mes complexes unitaires de de@ n est la variCtP des polynbmes ayant au mains une racine double. Pour n assez.grand, l’ensemble des racines d’un polynbme peut avoir des singularit& plus complexes. Le discriminant est strati@ par les variPt& de polyniimes ayant des singularit& don&es. Nous dtudions les groupes de cohomologie du compl.&mentaire de strates du discriminant et nous construisons une dkomposition cellulaire de la n-Pme puissance symktrique de 63 (Ccompatible )) avec la strat@cation. En utilisant cette d&composition, on peut calculer les premiers groupes de cohomologie. 0 Acadkmie des SciencesElsevier,
Paris
Version
frangaise
abrt?g&e
Soit P” I’ensemble des polyn6mes unitaires de degr6 n B coefficients complexes. Un polyn6me gBnCriquede degrC n admet n racines simples. Le discriminant de P” est la vari&tk des polyn6mes de Pn ayant une racine double. Le discriminant est une hypersurface algkbrique complexe singulihre. Le lieu singulier du discriminant est la &union de deux vari&& irrkductibles : la caustique (ensemble des polyn6mes ayant une racine triple) et la strate de Maxwell (ensemble des polyn6mes ayant deux Note pr&entCe 0764.4442/98/03270665
par Vladimir
ARNOL'D.
0 AcadCmie
des Sciences/Elsevier,
Paris
665
F. Napolitano
racines doubles). Ces variCtCs &ant elles-mCme singulibres, en continuant le mCme pro&d& nous obtenons une stratification naturelle du discriminant. Le but principal de cette Note est 1’Ctude de la cohomologie du complkmentaire des strates du discriminant. Pour toutes collections d’entiers positifs (al,. . . , aj) et (PI,. . . , &) tels que /31 > 02 > . . . > B”’ $3 ,f?j-1 > pj > 1, dkfinissons A,’ ’ ’ ’ comme l’ensemble des polynbmes complexes de degr6 a + xi=, ai,&, ayant ai racines de multiplicid &, pour 1 5 i 2 j. Quand il n’y a pas d’ambiguitk sur le degrk des poly&mes, l’ensemble A,“” “““,“’ est not6 simplement Ap~13...~p~‘. Ainsi, le discriminant est not6 AZ, la caustique A3 et la strate de Maxwell A2’. Une strate du discriminant de P” est une &union d’ensemble APT’ Y...Jj’T” c P”. Soit p E P” un polyn6me de racines ~1,. . . , z,. DCfinissons !I?@) E Pn+’ comme le polyn6me dont les racines sont 21,. . . , Z, et z,+l = maxlsi5n lzil i- 1. L’application !I! p&serve la multiplicitk des racines, done elle prkserve les strates du discriminant et leur complkmentaire. pa1 p 8”’ p-3 Soit ConsidCrons une strate A,’ ’ ’ 3 c P”. DCfinissons G,’ ’ ’ 3 = P” - A$’ ‘.““IJ. rk = 2(@k - 1) pour 1 5 k 5 j, R = c”,,, @rk et g le plus grand commun diviseur de , &. Nous prouvons l’analogue suivant des rksultats de [l] : I%,... pnl . . pai TH~ORI~ME 1. - Les groupes de cohomologie de G,’ ’ ’ ’ w!rijent les th&ort?mes de rkpe’tition, finitude et stabilite’ :
(1) Hi(Gp..‘tJ;~ Hi(Gy
Les isomorphismes
>...?0,“” ) = H”(Gf;’
,W)=O . ...I0,“”
)
pour
if0
pour
u>u’>(i+j+l-R)-----.
entre groupes de cohomologie
et
i#R-1,
sont induits par l’application
(2) 6 (9 - 1)
\Ir.
La suite exacte de Mayer-Vietoris
implique des rksultats semblables pour le complCmentaire d’une /3”’ ,.. p &union de strates irrkductibles A,’ ’ ’ ’ . Pour toute strate X de P”, notons x le compactifib par un point de X. DCfinissons Hci,(x) = HdimX-i(X, co) (dimX est le maximum des dimensions des composantes irrkductibles de X). La preuve du thCo&me 1 utilise la dualit d’Alexander : Hi(P” - X) = H(i+l+dimdy-2rL)(X), pour i > 0. Notons Sym”(C) la n-&me puissance symttrique de 43. L’application de Vikte et le thCor*me fondamental de l’alghbre impliquent P” N Sym”(C). Nous construisons une dkcomposition cellulaire de Sym”(C) compatible avec la stratification du discriminant (chaque strate est une &union de cellules). En utilisant cette dkcomposition et la dualit d’Alexander nous calculons quelques groupes de cohomologie (tableaux 1 et 2). Soit A = A$’ -p:’ c P”. Un polyn6me gCnt%ique p E A a exactement ui racines de multiplicitC ,8i pour 1 5 i 5 j, les a racines restantes Btant simples. Soit H un plan complexe de dimension 2n - dim A, passant par p, transversal B A en p. Soit I/ une petite boule ouverte cent&e en p. p,...,pqj Dtfinissons C, = H II V f~ A2 (pour V assez petit., B diffkomorphisme p&s, cet ensemble ne dCpend pas de p, H et V). L’ensemble C est appelC compl6ment local du discriminant au voisinnage de A. Notons Br(k) le groupe des tresses ci k brins (voir [l]). 666
Topology
of complements
of strata of the discriminant
of polynomials
/3”’ a07 2. - Le complement local du discriminant au voisinage de A,’ ’ ’ ’ est un espace d’Eilenberg--MaeLane K(a, 1). Le groupe r est un produit de groupes de tresses : p pa3 R = rI{&P k=lBr(/!?i). En particulier, le type d’homotopie de ‘C,’ ’ ’ ’ ne depend pas de a. THBOR~ME
Par exemple, le complement local C 2’ du discriminant au voisinage de la strate de Maxwell est homotope au tore. La preuve de ce thboreme est trb facile. Neanmoins, il peut C:treutile dans la resolution du probleme d’Arnol’d sur le quasi-resolvent du groupe des tresses ([2], probleme 2).
1. Introduction
and notations
Define P” as the set of unitary complex polynomials of degree n. A generic polynomial of degree n has n simple roots. The discriminant of P” is the variety of polynomials p E P” having at least one double root. The discriminant is a singular complex algebraic hypersurface. The singular locus of the discriminant is the union of two irreducible varieties: the caustic (set of polynomials having a triple root) and the Maxwell stratum (set of polynomials having two double roots). Since those varieties are themselves singular, we get a natural stratification of the discriminant. The main purpose of this paper is to obtain informations on the cohomology of complements of strata of the discriminant. For any collections (al,. . . , aj) and (,&, . . . , /3j) of positive integers such that /J1 > f12 > +, . > @j-l > flj > 1, define Afpl’.““”
ai/?i, having 0”’ pa3 ai roots of multiplicity pi, for 1 5 i < j. When there is no ambiguity on the degree, the set A,’ ’ ’ 3 is simply denoted APf’ >...;‘I’. Hence the discriminant is deno’ted by A2, the caustic by A3 and the Maxwell stratum by A2*. A stratum of the discriminant of P” is a union of sets A”;’ ,...lpI’ c P”. Let p E P” be a polynomial with roots ~1: , . . , z,. Define Xl’(p) E Pn+l as the polynomial whose roots are zl,. . . , zn and z,+~ = maxl
as the set of complex polynomials
1. - The cohomology groups of G,’
’ ’’
of degree a + &
satisjj the theorems of repetition, jniteness
and stability:
(1) Hi(G~;lJ?J)
=
()
for
if0
and
i#R-1,
(21
The isomorphismsbetween the cohomology groups are induced by the map P.
Mayer-Vietoris exact sequence implies similar results for the complement pa’ .,. p:j strata A,l ’ ’ 3 .
of a union of irreducible
667
F. Napolitano
For any stratum X of P” denote by x the one point compactification of X. Define Hci,(X) = HdimX-i(X, co) (dim X denotes the maximum of the dimension of the irreducible components of X). Theorem 1 is proved using Alexander duality: Hi(P” - X) = Hci+l+dimX-2n)(J?), for i > 0. Denote by Sym”(C) the n-th symetric power of 43. By ‘Vieta map and the fundamental theorem of algebra P” N Sym”(C). In Section 3 we construct a cellular decomposition of Sym”(C) compatible with the stratification of the discriminant (any stratum is al union of cells). Using this decomposition and Alexander duality we can compute the cohomology of the complement of any stratum (see Tables 1 and 2). Table 1. - Homology of the Maxwell stratum: H(,) (AZ’). Tableau
1. - Homologie
de la strate de A4axweli:
H(i)
(AZ’).
32 ). Table 2. - Homology of A3,‘: lHc;)(A,’ 32 ). Tableau 2. - Homologie de A3s2: Hc;)(A,’
a\i
0
8” . $3
Consider a stratum A = A,’ ’ ’ 3 c P”. A generic polynomial p in A has exactly ai roots of multiplicity ,& for 1 5 i 2 j, the other a roots being simple. Consider a complex plane H of dimension 2n - dim A through p, transversal to A in p. Let V a small open ball centred in p. Define )$’ >...JqJ = H n V n A2 (sets associated with different p, V and H are diffeomorphic). The set C is the local complement of the discriminant in a neighbourhood of A. Denote by Br(k) the braid group on k strings (see [l]). p”’ pal THEOREM 2. - The local complement of the discriminant in a neighbourhood of A,’ ’ ’ 3 is an Eilenberg-MacLane space K(r, l), where 7r is a product of braid groups: 7r = I$=,II&Br(Pi). In 0”’ pa3 particular, the homotopy type of C,’ ’ ’ ’ does not depend on a.
This theorem is very easy to prove; nevertheless, it could be useful in the solution of Amol’d’s problem on the quasi-resolvent of braid groups ([2], problem 2). The outline of the proof is as follows. First, consider a small neighbourhood V of p in P”. It is straightforward that V n A2 E Cdim * x (H n V n A”). Since the roots of polynomials are continuous functions of the coefficients, for V small enough, V n A2 is diffeomorphic to the product C” x II~=,II&P~* - A2. Since Pot - A2 is a K(K, 1) space with x equals to the braid group on /& strings (see [l]), the theorem follows. For example, the local complement C22 of the discriminant in a neighbourhood of the Maxwell stratum is homotopic to the torus. 668
Topology
2. Repetition,
finiteness
of complements
of strata of the discriminant
of polynomials
and stability
For any subset X of Pn, define *X = x \ {Znel (Z - 1)). Denote by S”#X its /c-fold suspension (see [l]). Suspension does not alter homology Hci,(SL#X) = H:ci,(X). Define A, = A:” “““,“‘. a > 0, the homologies of h, and A,-, are linked by the following exact sequence (see [l]):
H,i,(*Aa> ---)H(i,&>
4 H&a-l)
--+ H(i+l)(*Aa).
For
(4)
To the set A, we associate sets S, A and B. Define S = S”‘g*A$2”“‘“Y’. Define A (resp. B) as the union, over all integers X1, . . ,Xj,X such that 0 < X1, 0 5 A; 5 ai for 2 5 i 5 j, 0x1 p(~2-w I...,/p-3) 3 0 5 X I: a and Xl,& = X& + . . . + X,/3, + X, of sets c”‘#*A,l,’ ’ (resp. p(al+xl) (%-XL?) $5 >...I I *A 1 & ). We say that A is simple if A = 0. a-A Define @ : S + A by @(@I,. . . , z,~),P) = II&(Z - .z,)fil x p and @(cc) = (oo). The map Q establishes a homeomorphism of complements S \ A N *A \ .B. The degree of the restriction of Qr: $1 @(*z-h) >...,P,(a3-X3) B~~‘+~L)rp~2--X2),,,~,P(~3~~~) i cc”’ #*A,‘,’ ’ is greater than 1. fAa-A -xj)
LEMMA 1. - For a = qg + p and g the greatest common divisor of PI, . . . , ,O.j, the following equalities
hold: H(i,(*A,) Hc,,(*A,,
W) = 0
= 0 for 0 < p <: g,
(5)
if a > 0 or if A is not simple,
(6)
- (j - 1). (7) J’ Proof offormula (7). - Formula (7) is proved by induction ton j and, for each j, by induction on a1 +... + aj + a. The proof for j = 1 is given in [l]. For j > 1 and al + . . . + aj + a < &, it is straightforward. Define d = dim S - dim A. The map @ induces the following commutative diagram, where the horizontal lines are the exact sequences of the pairs (S, A) and (*A, B):
. -. H(i)(B) -
HCi) (‘A,)
= 0 for i 5 (rj - 1);:
H(i+d)(S) 1 H(i+d)(*A) -
H(i+d)(S/A) la* H(i+d)(*AlB) -
H(i+1)(4 I@* H&+1,(B)-
H(i+~+d)(s) ...
Since + induces a homeomorphism S\A -+* A\B, the vertical arrow in the middle is an isomorphism. By induction, Hti+d)(S) = 0 for i 5 (rj - l)g - (j - 2) - d. The above exact sequence implies H(i+,)(A) = H(t+d)(S/A) f or i 5 (r-j - 1): - (j - 1) - d. Hence we get the following exact sequence for i 5 (rj - l)g - (j - 1) - d: ’
H(i)(B) Suppose A = ca’#*A,Li
ph p-w
2
H(i+d,(*A) -
H(;+,,(A).
I.‘.> 3 p("j-x,)
(when A is the union of several components, one p(“l+~l),@(“2-b) ,...,pja3-‘j) should use Mayer-Vietoris exact sequence). Then B = *Aolx 2 . By induction H(,)(B) = 0 and Hci+,)(A) = 0 for i 5 (r-j - l)? - (j - 1). Since d = Z(~~=, X;(l - j$) + x(1 - k)) 2 2X(1 - k), H(;)(A) = 0 for i < (rj - l)g - (j - 1). El 669
F. Napolitano
Formulae (5) and (6) are proved similarly (see [I] for the case j = 1). Lemma 1, the exact sequence (4) and Alexander duality imply Theorem 1.
3. Cellular
decomposition
of Sym”(C)
To any point in Sym” (C) we associatesome index. The cellular decomposition of Sym” (C) consists of sets of points with the same index, and the point at infinity. Any stratum of the discriminant is a union of cells of this decomposition (see [3] where a cellular decomposition of P” - A2 is constructed). Consider a configuration [ E Sym”(C) of points of C. The function z H Rez takes m distinct values w1 < v2 < ... < u, on I. For any integer j, 1 5 j 5: m, the function z H Imz takes kj distinct values w1 < 202< . . . < w, on the configuration < n {z E 43 ; Rez = uj}. For 1 5 1 5 kj, define CE.~J = Card{z E [; Rez = uj, Imz = ZQ}. Define index(<) as the collection (cl,. . . , cm), where for any integer i, 15 i 5 m, ci = (cyj,~,. . . , a+,). The sum of all integers in the collections cl,. . . , c, is equal to n. Denote by e(cl, . . . , c,) the setsof configurations [ E Sym” (C) of index (cl, . . . , c,). LEMMA 2. - The space Sym”(C) can be representedas ajfinite CW-complex whose cells are various sets of configurations < E Sym”(C) with the same index and the point at injnity. Any stratum X of P”, is a subcomplex of Sym”(43). For any two collections of integers cl = (~1, _. . , ak11,c2 = (n , . . , n), denote by S(cl, c2) the set of permutations of the collection (al, _ . , CY~,yl,. _. , n) leaving invariant the order of n). For any g E S(cl, ~2) denote by E(O) the sign of the permutation Ql,..., ak (rev. YI,..., 0. For any collection of integers c = (~1,. . . , nk) and any i, 1 5 i 5 k - 1, define 3,(c) = (Ql,..., ai-l,cui+ai+l,ai+2,..., ag). Define ICI = k and Lj = Cllilj 1~1, 0 5 j 5 m. LEMMA
3. - The integral boundary operator in the cell complex Sym”(C) de(cl,.
Cj”=l(-l)“-l-l CTE1(-l)j+l+L”
C/?-‘(-1)”
. .,Gn)
=
e(cl,. . . ,~j-~,&(cj),cj+~,
1 E(g) ~ES(c,>CI+l)
is given by:
e(cl,. . . ,cj+l,(~,cj+z,.
. . . ,c,) . . ,c,),
Using lemmas 2 and 3 and Alexander duality, we can co.mputethe cohomology of the complement of any stratum. Acknowledgment. I thank V.I. Arnol’d, for posingthe problem, B. Shapiro and V.A. Vassiliev for their interest in this work.
References [l] Amol’d V.I., On some topological invariantsof algebraic functions,Trans. Moscow Math. Sot. 21 (1970) 30-52. [2] Amol’d V.I., On some problems in singularity theory, Proc. Indian Acad. Sci. 99 (1) (1981) 1-9. [3] Fuchs D.B., Cohomology of the braid group mod 2, Funct. Anal. Appl. 8 (1970).
670
SCrie
I, p. 665-670,
1998
Geometry)
Topology of complements of strata of the discriminant of polynomials Fabien , Ecole
NAPOLITANO normale
E-mail: (Repu
suptrieure,
45, rue d’Ulm,
75005 Paris,
France
[email protected] le
9 juin
Abstract.
1998,
accept6
le
18
septembre
1998)
The discriminantof the set of unitary complex polynomialsof degreen is the variety of polynomialshaving at leastone doubleroot. For n Xarge enough,the set of rootsof a polynomial may have more complicatedsingularities.The discriminantis stratified by the varietiesof polynomialshaving given singularities.We study the cohomology groups of the complementof any stratumand we constructa cellular decomposition of the symmetric n-th power of C “compatible” with the stratification. Using this decomposition,one can computeby hand the first cohomologygroups.0 AcadCmie des Sciences/Elsevier,Paris Topologie du complbment des polyn6mes
R&urn&
des strates da. discriminant
Le discriminant de l’ensemble des polyn6mes complexes unitaires de de@ n est la variCtP des polynbmes ayant au mains une racine double. Pour n assez.grand, l’ensemble des racines d’un polynbme peut avoir des singularit& plus complexes. Le discriminant est strati@ par les variPt& de polyniimes ayant des singularit& don&es. Nous dtudions les groupes de cohomologie du compl.&mentaire de strates du discriminant et nous construisons une dkomposition cellulaire de la n-Pme puissance symktrique de 63 (Ccompatible )) avec la strat@cation. En utilisant cette d&composition, on peut calculer les premiers groupes de cohomologie. 0 Acadkmie des SciencesElsevier,
Paris
Version
frangaise
abrt?g&e
Soit P” I’ensemble des polyn6mes unitaires de degr6 n B coefficients complexes. Un polyn6me gBnCriquede degrC n admet n racines simples. Le discriminant de P” est la vari&tk des polyn6mes de Pn ayant une racine double. Le discriminant est une hypersurface algkbrique complexe singulihre. Le lieu singulier du discriminant est la &union de deux vari&& irrkductibles : la caustique (ensemble des polyn6mes ayant une racine triple) et la strate de Maxwell (ensemble des polyn6mes ayant deux Note pr&entCe 0764.4442/98/03270665
par Vladimir
ARNOL'D.
0 AcadCmie
des Sciences/Elsevier,
Paris
665
F. Napolitano
racines doubles). Ces variCtCs &ant elles-mCme singulibres, en continuant le mCme pro&d& nous obtenons une stratification naturelle du discriminant. Le but principal de cette Note est 1’Ctude de la cohomologie du complkmentaire des strates du discriminant. Pour toutes collections d’entiers positifs (al,. . . , aj) et (PI,. . . , &) tels que /31 > 02 > . . . > B”’ $3 ,f?j-1 > pj > 1, dkfinissons A,’ ’ ’ ’ comme l’ensemble des polynbmes complexes de degr6 a + xi=, ai,&, ayant ai racines de multiplicid &, pour 1 5 i 2 j. Quand il n’y a pas d’ambiguitk sur le degrk des poly&mes, l’ensemble A,“” “““,“’ est not6 simplement Ap~13...~p~‘. Ainsi, le discriminant est not6 AZ, la caustique A3 et la strate de Maxwell A2’. Une strate du discriminant de P” est une &union d’ensemble APT’ Y...Jj’T” c P”. Soit p E P” un polyn6me de racines ~1,. . . , z,. DCfinissons !I?@) E Pn+’ comme le polyn6me dont les racines sont 21,. . . , Z, et z,+l = maxlsi5n lzil i- 1. L’application !I! p&serve la multiplicitk des racines, done elle prkserve les strates du discriminant et leur complkmentaire. pa1 p 8”’ p-3 Soit ConsidCrons une strate A,’ ’ ’ 3 c P”. DCfinissons G,’ ’ ’ 3 = P” - A$’ ‘.““IJ. rk = 2(@k - 1) pour 1 5 k 5 j, R = c”,,, @rk et g le plus grand commun diviseur de , &. Nous prouvons l’analogue suivant des rksultats de [l] : I%,... pnl . . pai TH~ORI~ME 1. - Les groupes de cohomologie de G,’ ’ ’ ’ w!rijent les th&ort?mes de rkpe’tition, finitude et stabilite’ :
(1) Hi(Gp..‘tJ;~ Hi(Gy
Les isomorphismes
>...?0,“” ) = H”(Gf;’
,W)=O . ...I0,“”
)
pour
if0
pour
u>u’>(i+j+l-R)-----.
entre groupes de cohomologie
et
i#R-1,
sont induits par l’application
(2) 6 (9 - 1)
\Ir.
La suite exacte de Mayer-Vietoris
implique des rksultats semblables pour le complCmentaire d’une /3”’ ,.. p &union de strates irrkductibles A,’ ’ ’ ’ . Pour toute strate X de P”, notons x le compactifib par un point de X. DCfinissons Hci,(x) = HdimX-i(X, co) (dimX est le maximum des dimensions des composantes irrkductibles de X). La preuve du thCo&me 1 utilise la dualit d’Alexander : Hi(P” - X) = H(i+l+dimdy-2rL)(X), pour i > 0. Notons Sym”(C) la n-&me puissance symttrique de 43. L’application de Vikte et le thCor*me fondamental de l’alghbre impliquent P” N Sym”(C). Nous construisons une dkcomposition cellulaire de Sym”(C) compatible avec la stratification du discriminant (chaque strate est une &union de cellules). En utilisant cette dkcomposition et la dualit d’Alexander nous calculons quelques groupes de cohomologie (tableaux 1 et 2). Soit A = A$’ -p:’ c P”. Un polyn6me gCnt%ique p E A a exactement ui racines de multiplicitC ,8i pour 1 5 i 5 j, les a racines restantes Btant simples. Soit H un plan complexe de dimension 2n - dim A, passant par p, transversal B A en p. Soit I/ une petite boule ouverte cent&e en p. p,...,pqj Dtfinissons C, = H II V f~ A2 (pour V assez petit., B diffkomorphisme p&s, cet ensemble ne dCpend pas de p, H et V). L’ensemble C est appelC compl6ment local du discriminant au voisinnage de A. Notons Br(k) le groupe des tresses ci k brins (voir [l]). 666
Topology
of complements
of strata of the discriminant
of polynomials
/3”’ a07 2. - Le complement local du discriminant au voisinage de A,’ ’ ’ ’ est un espace d’Eilenberg--MaeLane K(a, 1). Le groupe r est un produit de groupes de tresses : p pa3 R = rI{&P k=lBr(/!?i). En particulier, le type d’homotopie de ‘C,’ ’ ’ ’ ne depend pas de a. THBOR~ME
Par exemple, le complement local C 2’ du discriminant au voisinage de la strate de Maxwell est homotope au tore. La preuve de ce thboreme est trb facile. Neanmoins, il peut C:treutile dans la resolution du probleme d’Arnol’d sur le quasi-resolvent du groupe des tresses ([2], probleme 2).
1. Introduction
and notations
Define P” as the set of unitary complex polynomials of degree n. A generic polynomial of degree n has n simple roots. The discriminant of P” is the variety of polynomials p E P” having at least one double root. The discriminant is a singular complex algebraic hypersurface. The singular locus of the discriminant is the union of two irreducible varieties: the caustic (set of polynomials having a triple root) and the Maxwell stratum (set of polynomials having two double roots). Since those varieties are themselves singular, we get a natural stratification of the discriminant. The main purpose of this paper is to obtain informations on the cohomology of complements of strata of the discriminant. For any collections (al,. . . , aj) and (,&, . . . , /3j) of positive integers such that /J1 > f12 > +, . > @j-l > flj > 1, define Afpl’.““”
ai/?i, having 0”’ pa3 ai roots of multiplicity pi, for 1 5 i < j. When there is no ambiguity on the degree, the set A,’ ’ ’ 3 is simply denoted APf’ >...;‘I’. Hence the discriminant is deno’ted by A2, the caustic by A3 and the Maxwell stratum by A2*. A stratum of the discriminant of P” is a union of sets A”;’ ,...lpI’ c P”. Let p E P” be a polynomial with roots ~1: , . . , z,. Define Xl’(p) E Pn+l as the polynomial whose roots are zl,. . . , zn and z,+~ = maxl
as the set of complex polynomials
1. - The cohomology groups of G,’
’ ’’
of degree a + &
satisjj the theorems of repetition, jniteness
and stability:
(1) Hi(G~;lJ?J)
=
()
for
if0
and
i#R-1,
(21
The isomorphismsbetween the cohomology groups are induced by the map P.
Mayer-Vietoris exact sequence implies similar results for the complement pa’ .,. p:j strata A,l ’ ’ 3 .
of a union of irreducible
667
F. Napolitano
For any stratum X of P” denote by x the one point compactification of X. Define Hci,(X) = HdimX-i(X, co) (dim X denotes the maximum of the dimension of the irreducible components of X). Theorem 1 is proved using Alexander duality: Hi(P” - X) = Hci+l+dimX-2n)(J?), for i > 0. Denote by Sym”(C) the n-th symetric power of 43. By ‘Vieta map and the fundamental theorem of algebra P” N Sym”(C). In Section 3 we construct a cellular decomposition of Sym”(C) compatible with the stratification of the discriminant (any stratum is al union of cells). Using this decomposition and Alexander duality we can compute the cohomology of the complement of any stratum (see Tables 1 and 2). Table 1. - Homology of the Maxwell stratum: H(,) (AZ’). Tableau
1. - Homologie
de la strate de A4axweli:
H(i)
(AZ’).
32 ). Table 2. - Homology of A3,‘: lHc;)(A,’ 32 ). Tableau 2. - Homologie de A3s2: Hc;)(A,’
a\i
0
8” . $3
Consider a stratum A = A,’ ’ ’ 3 c P”. A generic polynomial p in A has exactly ai roots of multiplicity ,& for 1 5 i 2 j, the other a roots being simple. Consider a complex plane H of dimension 2n - dim A through p, transversal to A in p. Let V a small open ball centred in p. Define )$’ >...JqJ = H n V n A2 (sets associated with different p, V and H are diffeomorphic). The set C is the local complement of the discriminant in a neighbourhood of A. Denote by Br(k) the braid group on k strings (see [l]). p”’ pal THEOREM 2. - The local complement of the discriminant in a neighbourhood of A,’ ’ ’ 3 is an Eilenberg-MacLane space K(r, l), where 7r is a product of braid groups: 7r = I$=,II&Br(Pi). In 0”’ pa3 particular, the homotopy type of C,’ ’ ’ ’ does not depend on a.
This theorem is very easy to prove; nevertheless, it could be useful in the solution of Amol’d’s problem on the quasi-resolvent of braid groups ([2], problem 2). The outline of the proof is as follows. First, consider a small neighbourhood V of p in P”. It is straightforward that V n A2 E Cdim * x (H n V n A”). Since the roots of polynomials are continuous functions of the coefficients, for V small enough, V n A2 is diffeomorphic to the product C” x II~=,II&P~* - A2. Since Pot - A2 is a K(K, 1) space with x equals to the braid group on /& strings (see [l]), the theorem follows. For example, the local complement C22 of the discriminant in a neighbourhood of the Maxwell stratum is homotopic to the torus. 668
Topology
2. Repetition,
finiteness
of complements
of strata of the discriminant
of polynomials
and stability
For any subset X of Pn, define *X = x \ {Znel (Z - 1)). Denote by S”#X its /c-fold suspension (see [l]). Suspension does not alter homology Hci,(SL#X) = H:ci,(X). Define A, = A:” “““,“‘. a > 0, the homologies of h, and A,-, are linked by the following exact sequence (see [l]):
H,i,(*Aa> ---)H(i,&>
4 H&a-l)
--+ H(i+l)(*Aa).
For
(4)
To the set A, we associate sets S, A and B. Define S = S”‘g*A$2”“‘“Y’. Define A (resp. B) as the union, over all integers X1, . . ,Xj,X such that 0 < X1, 0 5 A; 5 ai for 2 5 i 5 j, 0x1 p(~2-w I...,/p-3) 3 0 5 X I: a and Xl,& = X& + . . . + X,/3, + X, of sets c”‘#*A,l,’ ’ (resp. p(al+xl) (%-XL?) $5 >...I I *A 1 & ). We say that A is simple if A = 0. a-A Define @ : S + A by @(@I,. . . , z,~),P) = II&(Z - .z,)fil x p and @(cc) = (oo). The map Q establishes a homeomorphism of complements S \ A N *A \ .B. The degree of the restriction of Qr: $1 @(*z-h) >...,P,(a3-X3) B~~‘+~L)rp~2--X2),,,~,P(~3~~~) i cc”’ #*A,‘,’ ’ is greater than 1. fAa-A -xj)
LEMMA 1. - For a = qg + p and g the greatest common divisor of PI, . . . , ,O.j, the following equalities
hold: H(i,(*A,) Hc,,(*A,,
W) = 0
= 0 for 0 < p <: g,
(5)
if a > 0 or if A is not simple,
(6)
- (j - 1). (7) J’ Proof offormula (7). - Formula (7) is proved by induction ton j and, for each j, by induction on a1 +... + aj + a. The proof for j = 1 is given in [l]. For j > 1 and al + . . . + aj + a < &, it is straightforward. Define d = dim S - dim A. The map @ induces the following commutative diagram, where the horizontal lines are the exact sequences of the pairs (S, A) and (*A, B):
. -. H(i)(B) -
HCi) (‘A,)
= 0 for i 5 (rj - 1);:
H(i+d)(S) 1 H(i+d)(*A) -
H(i+d)(S/A) la* H(i+d)(*AlB) -
H(i+1)(4 I@* H&+1,(B)-
H(i+~+d)(s) ...
Since + induces a homeomorphism S\A -+* A\B, the vertical arrow in the middle is an isomorphism. By induction, Hti+d)(S) = 0 for i 5 (rj - l)g - (j - 2) - d. The above exact sequence implies H(i+,)(A) = H(t+d)(S/A) f or i 5 (r-j - 1): - (j - 1) - d. Hence we get the following exact sequence for i 5 (rj - l)g - (j - 1) - d: ’
H(i)(B) Suppose A = ca’#*A,Li
ph p-w
2
H(i+d,(*A) -
H(;+,,(A).
I.‘.> 3 p("j-x,)
(when A is the union of several components, one p(“l+~l),@(“2-b) ,...,pja3-‘j) should use Mayer-Vietoris exact sequence). Then B = *Aolx 2 . By induction H(,)(B) = 0 and Hci+,)(A) = 0 for i 5 (r-j - l)? - (j - 1). Since d = Z(~~=, X;(l - j$) + x(1 - k)) 2 2X(1 - k), H(;)(A) = 0 for i < (rj - l)g - (j - 1). El 669
F. Napolitano
Formulae (5) and (6) are proved similarly (see [I] for the case j = 1). Lemma 1, the exact sequence (4) and Alexander duality imply Theorem 1.
3. Cellular
decomposition
of Sym”(C)
To any point in Sym” (C) we associatesome index. The cellular decomposition of Sym” (C) consists of sets of points with the same index, and the point at infinity. Any stratum of the discriminant is a union of cells of this decomposition (see [3] where a cellular decomposition of P” - A2 is constructed). Consider a configuration [ E Sym”(C) of points of C. The function z H Rez takes m distinct values w1 < v2 < ... < u, on I. For any integer j, 1 5 j 5: m, the function z H Imz takes kj distinct values w1 < 202< . . . < w, on the configuration < n {z E 43 ; Rez = uj}. For 1 5 1 5 kj, define CE.~J = Card{z E [; Rez = uj, Imz = ZQ}. Define index(<) as the collection (cl,. . . , cm), where for any integer i, 15 i 5 m, ci = (cyj,~,. . . , a+,). The sum of all integers in the collections cl,. . . , c, is equal to n. Denote by e(cl, . . . , c,) the setsof configurations [ E Sym” (C) of index (cl, . . . , c,). LEMMA 2. - The space Sym”(C) can be representedas ajfinite CW-complex whose cells are various sets of configurations < E Sym”(C) with the same index and the point at injnity. Any stratum X of P”, is a subcomplex of Sym”(43). For any two collections of integers cl = (~1, _. . , ak11,c2 = (n , . . , n), denote by S(cl, c2) the set of permutations of the collection (al, _ . , CY~,yl,. _. , n) leaving invariant the order of n). For any g E S(cl, ~2) denote by E(O) the sign of the permutation Ql,..., ak (rev. YI,..., 0. For any collection of integers c = (~1,. . . , nk) and any i, 1 5 i 5 k - 1, define 3,(c) = (Ql,..., ai-l,cui+ai+l,ai+2,..., ag). Define ICI = k and Lj = Cllilj 1~1, 0 5 j 5 m. LEMMA
3. - The integral boundary operator in the cell complex Sym”(C) de(cl,.
Cj”=l(-l)“-l-l CTE1(-l)j+l+L”
C/?-‘(-1)”
. .,Gn)
=
e(cl,. . . ,~j-~,&(cj),cj+~,
1 E(g) ~ES(c,>CI+l)
is given by:
e(cl,. . . ,cj+l,(~,cj+z,.
. . . ,c,) . . ,c,),
Using lemmas 2 and 3 and Alexander duality, we can co.mputethe cohomology of the complement of any stratum. Acknowledgment. I thank V.I. Arnol’d, for posingthe problem, B. Shapiro and V.A. Vassiliev for their interest in this work.
References [l] Amol’d V.I., On some topological invariantsof algebraic functions,Trans. Moscow Math. Sot. 21 (1970) 30-52. [2] Amol’d V.I., On some problems in singularity theory, Proc. Indian Acad. Sci. 99 (1) (1981) 1-9. [3] Fuchs D.B., Cohomology of the braid group mod 2, Funct. Anal. Appl. 8 (1970).
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