Author’s abstract: We give a concrete realization of the solution to Hilbert’s 17th problem for real positive definite forms and Becker’s generalization to higher even powers. Suppose \(p = p(x_1, \ldots, x_n)\) is a real positive definite form of degree \(m\). Then \(p\) is a sum of squares of rational functions with denominator \((\sum x^2_i)^r\) for sufficiently large \(r\). This generalizes a theorem of Pólya for real even positive definite forms. The proof gives substantially more explicit information.
Let \(\varepsilon (p) = \inf \{p(u) : u \in S^{n - 1}\}/ \sup \{p(u) : u \in S^{n - 1}\}\), a measure of how close \(p\) is to having a nontrivial zero. If \(r \geq {nm(m - 1) \over (4 \log 2) \varepsilon (p)} - {n + m \over 2}\), then \(p \cdot (\sum x^2_i)^r\) is a nonnegative linear combination of a universal set of \((m + 2r)\)-th powers of linear forms in \(\mathbb{Q} [x_1, \ldots, x_n]\). If \(p \in K [x_1, \ldots, x_n]\) for a field \(K \subset \mathbb{R}\), then the linear combination has coefficients in \(K\). Further, if \(2k \mid m\), then \(p\) is a sum of \(2k\)- th powers of rational functions whose denominators are a suitable power of \(\sum x^2_i\). If \(p\) and \(q\) are both positive definite and \(\deg (p) - \deg (q)\) is a multiple of \(2k\), then the rational function \(p/q\) is a sum of \(2k\)-th powers of rational functions whose denominators are a suitable product of powers of \(q\) and \(\sum x^2_i\).