Lower bounds for polynomials of many variables

Abstract

A polynomial P(ξ) = P(ξ₁,..., ξ_n) is said to be almost hypoelliptic if all its derivatives D^νP(ξ) can be estimated from above by P(ξ) (see [16]). By a theorem of Seidenberg-Tarski it follows that for each polynomial P(ξ) satisfying the condition P(ξ) > 0 for all ξ ∈ Rⁿ, there exist numbers σ > 0 and T ∈ R¹ such that P(ξ) ≥ σ(1 + |ξ|)^T for all ξ ∈ Rⁿ. The greatest of numbers T satisfying this condition, denoted by ST(P), is called Seidenberg-Tarski number of polynomial P. It is known that if, in addition, P ∈ I_n, that is, |P(ξ)| → ∞ as |ξ| → ∞, then T = T(P) > 0. In this paper, for a class of almost hypoelliptic polynomials of n (≥ 2) variables we find a sufficient condition for ST(P) ≥ 1. Moreover, in the case n = 2, we prove that ST(P) ≥ 1 for any almost hypoelliptic polynomial P ∈ I₂.