Effective aspects of positive semi-definite real and complex polynomials

Abstract

The question of whether a positive semidefinite polynomial can be written as a sum of squares of rational functions was posed by Hilbert in the 1900s, and this question, together with some variants regarding positivity of polynomials have been of interest to many. Polya gave a constructive proof with certain conditions and we are interested in several related questions to Polya's theorem as well. We conduct a survey of results on the relations among certain subsets of real and complex positive semi-definite polynomials which are relevant to the above questions. In particular, we determine the minimum degree at which we have strict inclusion for number of variables up to 4, and collate them in tabular form. We also modify existing results of Reznick for effective aspects of real-valued bihomogeneous positive definite polynomials. Lastly, we obtain necessary as well as sufficient conditions for Polya semi-stability of positive semi-definite polynomials with effective estimates.