Irreducibility Criterion For Quasi-Ordinary Polynomials

Abstract

We give a criterion for a quasi ordinary polynomial to be locally irreducible. The criterion uses the notion of approximate roots and that of generalized Newton polygons. It generalizes the one given by Abhyankar for algebraic plane curves. Introduction Let K be an algebraically closed field of characteristic zero, and let R = K[[x1, . . . , xe]] = K[[x]] be the ring of formal power series in x1, . . . , xe over K. Let f = y+a1(x)y + . . .+an(x) be a nonzero polynomial of R[y], and suppose that f is irreducible in R[y]. Suppose that e = 1 and let g be a nonzero polynomial of R[y], then define the intersection multiplicity of f with g, denoted int(f, g), to be the x-order of the y resultant of f and g. The set of int(f, g), g ∈ R[y], defines a semigroup, denoted Γ(f). It is will known that a set of generators of Γ(f) can be computed from polynomials having the maximal contact with f (see [1] and [6]), namely, there exist g1, . . . , gh such that n, int(f, g1), . . . , int(f, gh) generate Γ(f) and for all 1 ≤ k ≤ h, the Newton-Puiseux expansion of gk coincides with that of f until a characteristic exponent of f . In [1], Abhyankar introduced a special set of polynomials called the approximate roots of f . These polynomials have the advantage that they can be calculated from the equation of f by using the Tschirnhausen transform. Suppose that e ≥ 2 and that the discriminant of f is of the form x1 1 . . . . .x Ne e .u(x1, . . . , xe), where u is a unit in K[[x]] (such a polynomial is called quasi-ordinary polynomial). By Abhyankar-Jung Theorem, the roots of f(x1, . . . , xe, y) = 0 are all in K[[x 1 n 1 , . . . , x 1 n e ]], in particular there exists a power series y(t1, . . . , te) = ∑ p cpt p1 1 . . . . .t pe e ∈ K[[t1, . . . , te]] such that f(t1 , . . . , t n e , y(t1, . . . , te)) = 0 and the other roots of f(t n 1 , . . . , t n e , y) = 0 are the conjugates of y(t1, . . . , te) with respect to the nth roots of unity in K. Given a polynomial g of R[y], we define the order of g to be the leading exponent with respect to the lexicographical order of the smallest homogeneous component of g(t1 , . . . , t n e , y(t1, . . . , te)). The set of orders of polynomials of R[y] defines a semigroup. In this paper we first prove that the canonical basis of (nZ) with the set of orders of the approximate roots of f generate the semigroup of f , then we give, using these approximate roots and the notion of generalized Newton polygons, a criterion for a quasi-ordinary polynomial to be irreducible. Note that if e = 1, then f is quasi-ordinary, in particular our results generalize those of Abhyankar (see [1] and [3]). The paper is organized as follows: in Section 1 we introduce the notion of approximate roots of a polynomial in one variable over a commutative ring with unity. In Section 2 we show how to associate a semigroup with an irreducible quasi-ordinary polynomial of R[y]. In Section 3 we introduce the notion of pseudo roots of a quasi-ordinary polynomial f then we prove that the orders of these polynomials together with the canonical basis of (nZ) give a set of generators of the semigroup of f . This result remains true if we replace the pseudo roots of f by its set of approximate roots. This is what we prove in Section 4. Sections 5 and 6 are devoted to the irreducibility criterion: in Section 5 we introduce the notion of generalized Newton polygon, and we define ∗2000 Mathematical Subject Classification: 32S25, 32S70. During the development of this work, the author visited the Department of Mathematics at the American University of Beirut, Lebanon. He would like to thank that institution for hospitality and support. He also would like to think the Center for Advanced Mathematical Sciences-CAMS for offering access to many facilities. †Universite d’Angers, Mathematiques, 49045 Angers cedex 01, France, e-mail:assi@univ-angers.fr