Arithmetic Multivariate Descartes' Rule

Details Arithmetic Multivariate Descartes' Rule Arithmetic Multivariate Descartes' Rule

2001-10-31

arxiv.org/abs/math/0110327v3

Mathematics

Number Theory

27 pages, needs svjour.cls and svinvmat.clo (both included) to compile. Maple code to verify computations included. This versi

Scientific paper

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.

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