Finiteness for Arithmetic Fewnomial Systems
Abstract
Description
Suppose L is any finite algebraic extension of either the ordinary rational numbers or the p-adic rational numbers. Also let g_1,...,g_k be polynomials in n variables, with coefficients in L, such that the total number of monomial terms appearing in at least one g_i is exactly m. We prove that the maximum number of isolated roots of G:=(g_1,...,g_k) in L^n is finite and depends solely on (m,n,L), i.e., is independent of the degrees of the g_i. We thus obtain an arithmetic analogue of Khovanski's Theorem on Fewnomials, extending earlier work of Denef, Van den Dries, Lipshitz, and Lenstra.
6 pages; 1 figure (file=5adic.ps); needs my mildly hacked versions of amsart.cls and jams-l.cls, which are included. This is the final version and removes all ramification hypotheses, quotes some more explicit bounds for the benefit of the reader, streamlines some proofs, and corrects some more typos
6 pages; 1 figure (file=5adic.ps); needs my mildly hacked versions of amsart.cls and jams-l.cls, which are included. This is the final version and removes all ramification hypotheses, quotes some more explicit bounds for the benefit of the reader, streamlines some proofs, and corrects some more typos