Arithmetic Multivariate Descartes' Rule

Abstract

Let L be any number field or $\mathfrak{p}$-adic field and consider F:=(f_1,...,f_k) where f_i is in L[x_1,...,x_n]\{0} for all i and there are exactly m distinct exponent vectors appearing in f_1,...,f_k. We prove that F has no more than 1+(cmn(m-1)^2 log m)^n geometrically isolated roots in L^n, where c is an explicit and effectively computable constant depending only on L. This gives a significantly sharper arithmetic analogue of Khovanski's Theorem on Fewnomials and a higher-dimensional generalization of an earlier result of Hendrik W. Lenstra, Jr. for the case of a single univariate polynomial. We also present some further refinements of our new bounds and briefly discuss the complexity of finding isolated rational roots.