Additive Complexity and the Roots of Polynomials Over Number Fields and p-adic Fields

Abstract

Consider any nonzero univariate polynomial with rational coefficients, presented as an elementary algebraic expression (using only integer exponents). Letting sigma(f) denotes the additive complexity of f, we show that the number of rational roots of f is no more than 15 + sigma(f)^2 (24.01)^{sigma(f)} sigma(f)!. This provides a sharper arithmetic analogue of earlier results of Dima Grigoriev and Jean-Jacques Risler, which gave a bound of C^{sigma(f)^2} for the number of real roots of f, for some constant C with 1<C<32. We extend our new bound to arbitrary finite extensions of the ordinary or p-adic rationals, roots of bounded degree over a number field, and geometrically isolated roots of multivariate polynomial systems. We thus extend earlier bounds of Hendrik W. Lenstra, Jr. and the author to encodings more efficient than monomial expansions. We also mention a connection to complexity theory and note that our bounds hold for a broader class of fields.