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(m1)^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 higherdimensional 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.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 October 2001
 arXiv:
 arXiv:math/0110327
 Bibcode:
 2001math.....10327R
 Keywords:

 Mathematics  Number Theory;
 Mathematics  Algebraic Geometry;
 Primary: 11G25;
 Secondary: 11G35;
 14G20
 EPrint:
 27 pages, needs svjour.cls and svinvmat.clo (both included) to compile. Maple code to verify computations included. This version removes a factor of n^n from the main bounds and includes extra discussion on additive complexity and the complexity of finding isolated rational roots. Also, more typos are corrected, and the numerical bounds from the examples are improved considerably