Multiquadratic Sum-of-Squares Lower Bounds Imply VNC$^1$ $\neq$ VNP
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915646808260608 |
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| author | Rossman, Benjamin Zhu, Davidson |
| author_facet | Rossman, Benjamin Zhu, Davidson |
| contents | The \emph{sum-of-squares (SoS) complexity} of a $d$-multiquadratic polynomial $f$ (quadratic in each of $d$ blocks of $n$ variables) is the minimum $s$ such that $f = \sum_{i=1}^s g_i^2$ with each $g_i$ $d$-multilinear. In the case $d=2$, Hrubeš, Wigderson and Yehudayoff (2011) showed that an $n^{1+Ω(1)}$ lower bound on the SoS complexity of explicit biquadratic polynomials implies an exponential lower bound for non-commutative arithmetic circuits. In this paper, we establish an analogous connection between general \emph{multiquadratic sum-of-squares} and \emph{commutative arithmetic formulas}. Specifically, we show that an $n^{d-o(\log d)}$ lower bound on the SoS complexity of explicit $d$-multiquadratic polynomials, for any $d = d(n)$ with $ω(1) \le d(n) \le O(\frac{\log n}{\log\log n})$, would separate the algebraic complexity classes VNC$^1$ and VNP. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_01227 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Multiquadratic Sum-of-Squares Lower Bounds Imply VNC$^1$ $\neq$ VNP Rossman, Benjamin Zhu, Davidson Computational Complexity The \emph{sum-of-squares (SoS) complexity} of a $d$-multiquadratic polynomial $f$ (quadratic in each of $d$ blocks of $n$ variables) is the minimum $s$ such that $f = \sum_{i=1}^s g_i^2$ with each $g_i$ $d$-multilinear. In the case $d=2$, Hrubeš, Wigderson and Yehudayoff (2011) showed that an $n^{1+Ω(1)}$ lower bound on the SoS complexity of explicit biquadratic polynomials implies an exponential lower bound for non-commutative arithmetic circuits. In this paper, we establish an analogous connection between general \emph{multiquadratic sum-of-squares} and \emph{commutative arithmetic formulas}. Specifically, we show that an $n^{d-o(\log d)}$ lower bound on the SoS complexity of explicit $d$-multiquadratic polynomials, for any $d = d(n)$ with $ω(1) \le d(n) \le O(\frac{\log n}{\log\log n})$, would separate the algebraic complexity classes VNC$^1$ and VNP. |
| title | Multiquadratic Sum-of-Squares Lower Bounds Imply VNC$^1$ $\neq$ VNP |
| topic | Computational Complexity |
| url | https://arxiv.org/abs/2512.01227 |