# On the Complexity of Hilbert’s 17th Problem

• Nikhil R. Devanur
• Richard J. Lipton
• Nisheeth K. Vishnoi
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3328)

## Abstract

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 2 n 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.

## Keywords

Rational Function Boolean Function Turing Machine Satisfying Assignment Arithmetic Circuit
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Authors and Affiliations

• Nikhil R. Devanur
• 1
• Richard J. Lipton
• 1
• Nisheeth K. Vishnoi
• 1
1. 1.College of ComputingGeorgia Institute of TechnologyAtlanta