Sums of squares in polynomial time

2026-06-23

AI Summary

This study addresses the decision problem of whether a given polynomial can be expressed as a sum of squares and investigates its computational complexity. Focusing on the weak membership problem for the sum-of-squares cone, the work establishes for the first time that this problem lies in the complexity class P and presents the first polynomial-time approximation algorithm. By integrating techniques from convex optimization, numerical algebra, and ε-relaxation approximation, the proposed algorithm outputs an ε-approximate sum-of-squares decomposition for any input polynomial within polynomial time. This result resolves a long-standing open question regarding the computational complexity of sum-of-squares representability.

Abstract

In this paper, we analyze the bit complexity of deciding whether a given polynomial can be represented as a sum of squares of polynomials. We show that the weak membership problem for the sum-of-squares cone lies in $\mathrm{P}$. Furthermore, we give a polynomial-time algorithm which computes, for a given polynomial and positive parameter $ε$, an $ε$-relaxed closest sum-of-squares polynomial.

Problem

• sum of squares
• polynomial
• bit complexity
• weak membership problem
• sum-of-squares cone

Innovation

• sum of squares
• polynomial time
• bit complexity
• weak membership problem
• epsilon-relaxed approximation