REDUCIBLE FIBERS OF POLYNOMIALS

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REDUCIBLE FIBERS OF POLYNOMIALS ARNAUD BODIN Abstract. We give an algorithm to find reducible fibers of a bi- variate polynomial. 1. Ruppert's equation, Gao's theorem Let F be any field of characteristic p. Let f ? F[x, y]. The aim of this note is to give a method to find the values c for which f ? c is reducible based on results of W. Ruppert and S. Gao, [Ru], [Ga]. We consider the following equation : (1) ∂ ∂y ( g f ? c ) = ∂ ∂x ( h f ? c ) , where g, h ? F[x, y] are unknows and f ? F[x, y], c ? F are given data. This equation can be “linearized” to the equivalent equation: (2) (f ? c) · ( ∂g ∂y ? ∂h ∂x ) = g · ∂f ∂y ? h · ∂f ∂x . Notice that this equation is linear in the coefficient of g and h. The bidegree of a polynomial g is deg g = (degx g, degy g). Let (m,n) be the bidgree of f . We impose the following restrictions: (3) deg g 6 (m ? 1, n), deg h 6 (m,n ? 1).

• stein fac- torization


• reducible fibers


• brianc¸on polynomial


• finding reducible


• pq ?


• polynomials via partial differential equations


• equation

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• Painlevé
• Équation

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• Publié par: profil-urra-2012
• Nombre de lectures: 9
• Langue: English

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REDUCIBLE FIBERS OF POLYNOMIALS

ARNAUD BODIN

Abstract.We give an algorithm to find reducible fibers of a bi- variate polynomial.

1.Ruppert’s equation, Gao’s theorem LetFbe any field of characteristicp. Letf∈F[x, yaim of]. The this note is to give a method to find the valuescfor whichf−cis reducible based on results of W. Ruppert and S. Gao, [Ru], [Ga]. We consider the following equation :    ∂ g∂ h (1) =, ∂y f−fc ∂x−c whereg, h∈F[x, y] are unknows andf∈F[x, y],c∈Fare given data. This equation can be “linearized” to the equivalent equation:   ∂g ∂h∂f ∂f (2) (f−c)∙ −=g∙ −h∙. ∂y ∂x∂y ∂x Notice that this equation is linear in the coefficient ofgandh. Thebidegreeof a polynomialgis degg= (degg,degg). Let(m, n) x y be the bidgree off. Weimpose the following restrictions: (3) degg6(m−1, n),degh6(m, n−1). The relation degg6(m−1, n) means degg6m−1 and degg6n. x y Note that relations (3) define a finite dimensional vector space for the coefficients ofgandh. We suppose thatfis not constant and that for allc∈F, gcd(f− ∂f c,Then for each) = 1.gthere is at most onehthat satisfies (2) and ∂x (3). Wedefine G={g∈F[x, y]|(2) and (3) holds for someh∈F[x, y]}. Gis a finite dimensional vector space and is at least one dimensional ∂f ∂f becauseg=∈G(forh= ). ∂x ∂y

Date: September 27, 2005.

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