Impossibility of extending Pólya's theorem to ``forms'' with arbitrary real exponents
The starting point of the paper under review is the well-known and important Theorem of Polya. Let \(F\) be a homogeneous polynomial in \(n\) variables such that \(F>0\) on \(\mathbb{R}_{\geq 0}\setminus{\{0\}}\). Then the following holds. Strong form of Polya's Theorem: For all sufficiently large \(p\in\mathbb{N}\) the homogeneous polynomial \((X_1+\ldots+X_n)^p F(X_1,\ldots,X_n)\) has strictly positive coefficients (which means that the coefficients at the monomials of the appropriate homogeneous degree are strictly positive). Weak form of Polya's Theorem: \(F\) is the quotient of two homogeneous polynomials with every coefficient strictly positive. The author discusses the extension of the results above to homogeneous generalized polynomials with arbitrary real exponents. He shows the following: - The strong form of Polya's Theorem does not extend to generalized polynomials with non-integer exponents. - The weak form of Polya's Theorem extends to generalized polynomials with rational exponents. - The weak form of Polya's Theorem does not extend to generalized polynomials with arbitrary real exponents. The paper is carefully written and very detailed. A lot of information about the background and the literature is given.
scientific article; zbMATH DE number 5354969