On the Infinite Increase of a Class of Polynomials

摘要

I-n (or (I) over tilde (n), respectively) denotes the set of polynomials in n variables P(xi) = P(xi(1), xi(2),..., xi(n)) with real coefficients such that vertical bar P(xi)vertical bar - infinity as vertical bar xi vertical bar - infinity (respectively, (P) over tilde(xi) := Sigma(nu)vertical bar(DP)-P-nu(xi)vertical bar - infinity as vertical bar xi vertical bar - infinity. The necessary and sufficient conditions for P is an element of I-n (or for P is an element of I-n, respectively) are obtained in terms of degree m, the geometrical structure of the Newton polyhedron R(P) of polynomial P, and the multiplicity of zeros of its main homogeneous subpolynomial P-m. The geometrically illustrative sufficient conditions are found for R is an element of(I) over tilde (3) for generalized homogeneous polynomials R in three variables.