Sums of squares in polynomial time
arXiv:2606.251187.9
Predicted impact top 35% in OC · last 90 daysOriginality Highly original
AI Analysis
This resolves the computational complexity of a fundamental problem in polynomial optimization, providing a polynomial-time algorithm for a previously open question.
The paper shows that the weak membership problem for the sum-of-squares cone is in P, and provides a polynomial-time algorithm to compute an ε-relaxed closest sum-of-squares polynomial.
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.