Polynomials over ordered fields
There is a theorem, published in 1928 by Pólya, about the representation of any real form \(F\) which is positive on the region \(\{(x_1,\ldots,x_n)\in \mathbb R^n : \text{\;all\;} x_j\geq0 \text{\;and\;} \sum x_j>0\}\) as a quotient \(G/H\), where \(G\) is a form all whose coefficients are positive and \(H=(x_1+\ldots+x_n)^p\), for some \(p\in\mathbb N\). Written as \(G/H\), the positivity of \(F\) is evident. The paper under review can be considered as a vast generalization of Pólya's theorem. In the first place, the real field is replaced by any ordered field \(K\). Let \(\overline K\) be the real closure of \(K\). Let \(X=(X_1,\ldots, X_n)\) be indeterminates. For any \(F\in K[X]\), let \(| F| \) denote the polynomial whose coefficients are the absolute values of the coefficients of \(F\). The author proves the following theorem (Theorem 3.1): given a form \(0\neq F\) in \(K[X]\), there exist forms \(G,H\) in \(K[X]\) all whose coefficients are positive so that \(FH=G\) if and only if there exists a rational \(\epsilon>0\) such that \(\epsilon | F| (x)< F(x)\), for all \(x\in \{(x_1,\ldots,x_n)\in {\overline K}^n : \text{\;all\;} x_j\geq0 \text{\;and\;} \sum x_j>0\}\). This result follows from the paper's main theorem: given a polynomial \(0\neq P\) in \(K[X]\), there exists a polynomial \(0\neq H\) in \(K[X]\) such that all the coefficients of \(FH\) are non-negative if and only if there exists a rational \(\epsilon>0\) such that \(\epsilon | F| (x)< F(x)\), for all \(x\in \{(x_1,\ldots,x_n)\in {\overline K}^n : \text{\;all\;} x_j\geq0 \}\). Moreover, if such an \(H\) exists, it can be chosen having all coefficients non-negative or having all coefficients non-positive. The techniques used in the proofs come from the theories of ordered fields and of convex sets and polytopes. As the author points out, Pólya's method does not apply to non-Archimedean fields. In addition to this paper, there are several papers about Pólya's theorem which might be interesting to the reader (none of them appear in the bibliography), such as [\textit{G.-X. Zeng}, ``Ordered fields satisfying Pólya's theorem'', Proc. Am. Math. Soc. 133, No. 10, 2921--2926 (2005; Zbl 1105.12002)] and papers cited therein.
scientific article; zbMATH DE number 5129955