Sums of squares in polynomial time

Nikolas Gärtner; Victor Magron; Frank Vallentin

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arXiv · 2606.25118

Sums of squares in polynomial time

Nikolas Gärtner·Victor Magron·Frank Vallentin

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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 $\epsilon$, an $\epsilon$-relaxed closest sum-of-squares polynomial.

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Nikolas Gärtner, Victor Magron, Frank Vallentin. 2026-06-23. Sums of squares in polynomial time. https://arxiv.org/abs/2606.25118

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