Lower Bounds for Polynomials of Many Variables

摘要

A polynomial P(xi) = P(xi(1), ..., xi(n)) is said to be almost hypoelliptic if all its derivatives D.P(xi) can be estimated from above by P(xi) (see [16]). By a theorem of Seidenberg-Tarski it follows that for each polynomial P(xi) satisfying the condition P(xi) 0 for all xi is an element of R-n, there exist numbers sigma 0 and T is an element of R-1 such that P(xi) = sigma(1 + vertical bar xi vertical bar)(T) for all xi is an element of R-n. 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 is an element of I-n, that is, vertical bar P(xi)vertical bar - infinity as vertical bar xi vertical bar - infinity, 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 is an element of I-2.