Arithmetic Multivariate Descartes Rule
Abstract: Rene Descartes stated no later than june of 1637 that any real univariate polynomial with exactly m monomial terms has at most m-1 positive roots --- an upper bound totally independent of the degree. Finding a sharp generalization to multivariate polynomial systems has elluded us so far, so we report on a surprising advance in a slightly different direction: An arithmetic multivariate analogue of Descartes' bound, which bounds the number of geometrically isolated roots over any finite algebraic extension of the ordinary or p-adic rationals, and is asymptotically near optimal: The upper bound is
1 + [C n (m-n)^3 log m]^n
where m is the total number of distinct exponent vectors, n is the number of variables, and C is a constant depending only on the underlying field.
This result yields higher-dimensional generalizations of earlier results of Hendrik W. Lenstra, Jr. for the case of a single univariate polynomial, provides a sharper analogue of an earlier result of Khovanski over the real numbers, and greatly simplifies earlier work of Denef, Lipshitz, and van den Dries which involved rigid analytic geometry and model theory. We thus obtain the foundations for an arithmetic analogue of Khovanski's theory of fewnomials.
No background in number theory or algebraic geometry is assumed.