An Effective Łojasiewicz Inequality for Real Polynomials

Abstract

Let H be the supremum of finitely many real polynomials of degree d and assume that H has a strict local minimum at 0. We prove a Łojasiewicz-type inequality $H(x_1,...,x_n) > ||(x_1,...,x_n)||^s$ where s depends only on d and n. This implies a similar inequality where $(x_1,...,x_n)$ runs through the points of a semi-algebraic set.