Does co-NP have short interactive proofs?
Information Processing Letters
Greatest common divisors of polynomials given by straight-line programs
Journal of the ACM (JACM)
Probabilistic checking of proofs: a new characterization of NP
Journal of the ACM (JACM)
Probabilistic Algorithms for Deciding Equivalence of Straight-Line Programs
Journal of the ACM (JACM)
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
Hilbert's Twenty-Fourth Problem
Journal of Automated Reasoning
The complexity of theorem-proving procedures
STOC '71 Proceedings of the third annual ACM symposium on Theory of computing
Probabilistic checking of proofs; a new characterization of NP
SFCS '92 Proceedings of the 33rd Annual Symposium on Foundations of Computer Science
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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 numberof rational functions needed to represent any non-negative n-variate, degree dpolynomial? 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 complexityof the sum of squares representation (as rational functions) of a non-negative, n-variate, degree dpolynomial? 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.