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Irreducibility criterion for quasi-ordinary polynomials ∗
| Content Provider | Semantic Scholar |
|---|---|
| Author | Assi, Abdallah |
| Copyright Year | 2009 |
| Abstract | 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 n +a1(x)y n−1 + . . .+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]). |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | http://arxiv.org/pdf/0904.4413v1.pdf |
| Language | English |
| Access Restriction | Open |
| Subject Keyword | Approximation algorithm Arabic numeral 0 Discriminant Eisenstein's criterion Existential quantification Immunostimulating conjugate (antigen) Irreducibility Lexicographical order Lexicography Maximal set Newton Newton's method Newton-X Norepinephrine Plant Roots Polynomial Resultant Root of unity Tellurium multiplicity |
| Content Type | Text |
| Resource Type | Article |
