Finiteness for Arithmetic Fewnomial Systems

Details Finiteness for Arithmetic Fewnomial Systems Finiteness for Arithmetic Fewnomial Systems

2000-10-27

arxiv.org/abs/math/0010260v3

Mathematics

Number Theory

6 pages; 1 figure (file=5adic.ps); needs my mildly hacked versions of amsart.cls and jams-l.cls, which are included. This is t

Scientific paper

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.

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