On the Complexity of Hilbert’s 17th Problem

Hilbert posed the following problem as the 17th in the list of 23 problems in his famous 1900 lecture: Given a multivariate polynomial that takes only non-negative values over the reals, can it be represented as a sum of squares of rational functions? In 1927, E. Artin gave an affirmative answer to this question. His result guaranteed the existence of such a finite representation and raised the following important question: What is the minimum number of rational functions needed to represent any non-negative n-variate, degree d polynomial? In 1967, Pfister proved that any n-variate non-negative polynomial over the reals can be written as sum of squares of at most 2n rational functions. In spite of a considerable effort by mathematicians for over 75 years, it is not known whether n + 2 rational functions are sufficient! In lieu of the lack of progress towards the resolution of this question, we initiate the study of Hilbert’s 17th problem from the point of view of Computational Complexity. In this setting, the following question is a natural relaxation: What is the descriptive complexity of the sum of squares representation (as rational functions) of a non-negative, n-variate, degree d polynomial? We consider arithmetic circuits as a natural representation of rational functions. We are able to show, assuming a standard conjecture in complexity theory, that it is impossible that every non-negative, n-variate, degree four polynomial can be represented as a sum of squares of a small (polynomial in n) number of rational functions, each of which has a small size arithmetic circuit (over the rationals) computing it. Our result points to the direction that it is unlikely that every non-negative, n-variate polynomial over the reals can be written as a sum of squares of a polynomial (in n) number of rational functions. Further, relating to standard (and believed to be hard to prove) complexity-theoretic conjectures sheds some light on why it has been difficult for mathematicians to close the n + 2 and 2n gap. We hope that our line of work will play an important role in the resolution of this question.