Polynomials over ordered fields

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]F(X1,…,Xn)∈R[X1,…,Xn] is a form (i.e., a homogeneous polynomial) s.t. F(x1,…,xn)>0∀x1≥0,…,xn≥0 with ∑xj>0∑xj>0, then F=G/HF=G/H, where G,HG,H are forms with all coefficients positive (i.e., every monomial of degree degGdegG or degHdegH appears in GG or HH, resp., with a coefficient that is strictly positive). In Pólya’s proof HH is chosen to be H=(X1+⋯+Xn)mH=(X1+⋯+Xn)m for some mm.At the end we give some applications, including a generalization of Pólya’s result to arbitrary ordered fields.