Polynomials over ordered fields |
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Authors: | Constantin N. Beli |
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Affiliation: | Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, RO-70700 Bucharest, Romania |
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Abstract: | In this paper we determine which polynomials over ordered fields have multiples with nonnegative coefficients and also which polynomials can be written as quotients of two polynomials with nonnegative coefficients. This problem is related to a result given by Pólya in [G.H. Hardy, J.E. Littlewood, G. Pólya, Inequalities, Cambridge University Press, Cambridge, England, 1952] (as a companion of Artin’s theorem) that asserts that if F(X1,…,Xn)∈R[X1,…,Xn] is a form (i.e., a homogeneous polynomial) s.t. with ∑xj>0, then F=G/H, where G,H are forms with all coefficients positive (i.e., every monomial of degree degG or degH appears in G or H, resp., with a coefficient that is strictly positive). In Pólya’s proof H is chosen to be H=(X1+?+Xn)m for some m.At the end we give some applications, including a generalization of Pólya’s result to arbitrary ordered fields. |
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Keywords: | 12D15 11E10 52B11 |
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