A product formula for minimal polynomials and degree bounds for inverses of polynomial automorphisms

Abstract

By means of Galois theory, we give a product formula for the minimal polynomial G of {fo, fl ... , fn} C K[xl, ... , xn] which contains n algebraically independent elements, where K is a field of characteristic zero. As an application of the product formula, we give a simple proof of Gabber's degree bound inequality for the inverse of a polynomial automorphism. 0. INTRODUCTION Let K be a field, and let {fo, ... , fn} C K[xl, ..., xn] contain n algebraically independent polynomials over K. Then there is a unique irreducible polynomial (up to a constant factor in K*) G(yo, ... , Yn) E K[y1, .. , Yn ] such that G(fo, ... , fn) = 0. We call this G the minimal polynomial of fo, ... , fn over K. It can be viewed as a natural generalization of the minimal polynomial of an algebraic element over a field K. Minimal polynomials are very useful for studying polynomial automorphisms, as well as birational maps. See, for instance, Yu [11, 12] and Li and Yu [3, 4]. In [3] and [12], two different effective algorithms for computing minimal polynomials are given, by means of Grobner bases and Generalized Characteristic Polynomials (GCP), respectively. The following theorem is well known. Theorem 0.1. Let a be algebraic over a field K and ma(x) be the minimal polynomial of a over K. Then d ma(x) = ]J(x a(')) i=1 where a0), ... , a (d) are all roots of the polynomial ma(x) in the algebraic closure of K(a) and deg(ma(x)) = d, the number of roots of ma (x) . Received by the editors December 22, 1992 and, in revised form, May 5, 1993; presented at AMS Special Session Geometry of Affine Space, Springfield, MO, March 20-21, 1992. 1991 Mathematics Subject Classification. Primary 12E05, 12F05, 12Y05, 13B05.