Lower bounds for polynomials of many variables |
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Authors: | V N Margaryan H G Ghazaryan |
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Institution: | 1.Russian-Armenian (Slavonic) University,Yerevan,Armenia;2.Institute of Mathematics NAS of Armenia,Yerevan,Armenia |
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Abstract: | A polynomial P(ξ) = P(ξ1,..., ξ n ) is said to be almost hypoelliptic if all its derivatives D ν P(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 ∈ R n , there exist numbers σ > 0 and T ∈ R1 such that P(ξ) ≥ σ(1 + |ξ|) T for all ξ ∈ 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 ∈ 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. |
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