Vol 52, No 5 (2017)

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.

The paper contains results on localization of singularities for the power series (in terms of their coefficients) on boundary arcs of the disk of convergence.

In this paper, we obtain recovery formulas for coefficients of multiple Franklin series by means of its sum, if the series satisfies the following conditions: 1) the square partial sums with indices 2^ν converge almost everywhere, 2) the majorant of partial sums with indices 2^ν satisfies some necessary condition.