NEWTON POLYGONS AND FAMILIES OF POLYNOMIALS

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Niveau: Secondaire, Lycée, Première
NEWTON POLYGONS AND FAMILIES OF POLYNOMIALS ARNAUD BODIN Abstract. We consider a continuous family (fs), s ? [0, 1] of complex polynomials in two variables with isolated singularities, that are Newton non-degenerate. We suppose that the Euler characteristic of a generic fiber is constant. We firstly prove that the set of critical values at infinity depends continuously on s, and secondly that the degree of the fs is constant (up to an algebraic automorphism of C2). 1. Introduction We consider a family (fs)s?[0,1] of complex polynomials in two variables with isolated singularities. We suppose that coefficients are continuous func- tions of s. For all s, there exists a finite bifurcation set B(s) such that the restriction of fs above C \ B(s) is a locally trivial fibration. It is known that B(s) = Baff (s) ? B∞(s), where Baff (s) is the set of affine critical val- ues, that is to say the image by fs of the critical points; B∞(s) is the set of critical values at infinity. For c /? B(s), the Euler characteristic verifies ?(f?1s (c)) = µ(s) + ?(s), where µ(s) is the affine Milnor number and ?(s) is the Milnor number at infinity.

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• newton polygon

Sujets

• Première
• Newton polygon

Informations

• Publié par: profil-ondu-2012
• Nombre de lectures: 15
• Langue: English

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NEWTON POLYGONS AND FAMILIES OF POLYNOMIALS

ARNAUD BODIN

Abstract.We consider a continuous family (fs),s∈[0,1] of complex
polynomials in two variables with isolated singularities, that are Newton
non-degenerate. Wesuppose that the Euler characteristic of a generic
fiber is constant.We firstly prove that the set of critical values at
infinity depends continuously ons, and secondly that the degree of the
2
fsis constant (up to an algebraic automorphism ofC).

1.Introduction
We consider a family (fs)s∈[0,1]of complex polynomials in two variables
with isolated singularities.We suppose that coefficients are continuous
functions ofs. Foralls, there exists a finitebifurcation setB(s) such that the
restriction offsaboveC\ B(s) is a locally trivial fibration.It is known
thatB(s) =Baff(s)∪ B∞(s), whereBaff(s) is the set ofaffine critical
values, that is to say the image byfsof the critical points;B∞(s) is the set
ofcritical values at infinity. Forc /∈ B(s), the Euler characteristic verifies
−1
χ(f(c)) =µ(s) +λ(s), whereµ(s) is theaffine Milnor numberandλ(s)
s
is theMilnor number at infinity.
We will be interested in families such that the sumµ(s) +λ(s) is constant.
These families are interesting in the view ofµ-constant type theorem, see
[HZ, HP, Ti, Bo, BT].We say that a multi-valued functions7→F(s) is
continuousif at each pointσ∈[0,1] and at each valuec(σ)∈F(σ) there is
a neighborhoodIofσsuch that for alls∈I, there existsc(s)∈F(s) near
c(σ).Fisclosed, if, for all pointsσ∈[0,1], for all sequencesc(s)∈F(s),
s6=σ, such thatc(s)→c(σ)∈Cass→σ, thenc(σ)∈F(σ). Itit is
well-known thats7→ Baff(sBut it) is a continuous multi-valued function.
is not necessarily closed:for examplefs(x, y) = (x−s)(xy−1), then for
s6= 0,Baff(s) ={0, s}butBaff(0) =∅.
We will prove thats7→ B∞(s) ands7→ B(s) are closed continuous
functions under some assumptions.
Theorem 1.Let(fs)s∈[0,1]be a family of complex polynomials such that
µ(s) +λ(s)is constant and such thatfsis (Newton) non-degenerate for
alls∈[0,1], then the multi-valued functions7→ B∞(s)is continuous and
closed.
Remark.isAs a corollary we get the answer to a question of D. Siersma:
it possible to find a family withµ(s) +λ(s) constant such thatλ(0)>0
(equivalentlyB∞(0)6=∅) andλ(s) = 0 (equivalentlyB∞=∅) fors∈
1