Universality of Polynomial Positivity and a Variant of Hilbert’s 17th Problem

We observe that the decision problem for the ∃ theory of real closed fields (RCF) is simply reducible to the decision problem for RCF over a connective-free ∀ language in which the only relation symbol is a strict inequality. In particular, every ∃ RCF sentence φ can be settled by deciding a proposition of the form “polynomial p (which is a sum of squares) takes on strictly positive values over the reals,” with p simply derived from φ. Motivated by this observation, we pose the goal of isolating a syntactic criterion characterising the positive definite (i.e., strictly positive) real polynomials. Such a criterion would be a strictly positive analogue to the fact that every positive semidefinite (i.e., nonnegative) real polynomial is a sum of squares of rational functions, as established by Artin’s positive solution to Hilbert’s 17th Problem. We then prove that every positive definite real polynomial is a ratio of a Real Nullstellensatz witness and a positive definite real polynomial. Finally, we conjecture that every positive definite real polynomial is a product of ratios of Real Nullstellensatz witnesses and examine an interesting ramification of this conjecture.