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A Subquadratic Upper Bound on Sum-Of-Squares Composition Formulas

Author Pavel Hrubeš

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Author Details

Pavel Hrubeš
  • Institute of Mathematics of ASCR, Prague, Czech Republic

Funding

  • Hrubeš, Pavel: This work was supported by Czech Science Foundation GAČR grant 19-27871X.

Cite As Get BibTex

Pavel Hrubeš. A Subquadratic Upper Bound on Sum-Of-Squares Composition Formulas. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 12:1-12:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.CCC.2024.12

Abstract

For every n, we construct a sum-of-squares identity (∑_{i=1}^n x_i²) (∑_{j=1}^n y_j²) = ∑_{k=1}^s f_k², where f_k are bilinear forms with complex coefficients and s = O(n^1.62). Previously, such a construction was known with s = O(n²/log n). The same bound holds over any field of positive characteristic.

Subject Classification

ACM Subject Classification
  • Theory of computation
Keywords
  • Sum-of-squares composition formulas
  • Hurwitz’s problem
  • non-commutative arithmetic circuit

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References

  1. Marco L. Carmosino, Russell Impagliazzo, Shachar Lovett, and Ivan Mihajlin. Hardness amplification for non-commutative arithmetic circuits. In Proceedings of the 33rd Computational Complexity Conference, CCC '18, 2018. Google Scholar
  2. A. Geramita and N. Pullman. A theorem of Hurwitz and Radon and orthogonal projective modules. Proceedings of The American Mathematical Society, 42:51-51, January 1974. URL: https://doi.org/10.1090/S0002-9939-1974-0332764-4.
  3. I. Haviv. On minrank and the Lovász Theta function. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, 2018. Google Scholar
  4. I. Haviv. Topological bounds on the dimension of orthogonal representations of graphs. European Journal of Combinatorics, 81:84-97, 2019. Google Scholar
  5. P. Hrubeš. On families of anticommuting matrices. Linear Algebra and Applications, 493:494-507, 2016. Google Scholar
  6. P. Hrubeš, A. Wigderson, and A. Yehudayoff. Non-commutative circuits and the sum of squares problem. In STOC' 10 Proceedings of the 42nd symposium on Theory of Computing, pages 667-676, 2010. Google Scholar
  7. P. Hrubeš, A. Wigderson, and A. Yehudayoff. An asymptotic bound on the composition number of integer sums of squares formulas. Canadian Mathematical Bulletin, 56:70-79, 2013. Google Scholar
  8. A. Hurwitz. Über die Komposition der quadratischen Formen von beliebigvielen Variabeln. Nach. Ges. der Wiss. Göttingen, pages 309-316, 1898. Google Scholar
  9. L. Lovász. On the Shannon capacity of a graph. IEEE Trans. Inform. Theory, 25(1):1-7, 1979. Google Scholar
  10. N. Nisan. Lower bounds for non-commutative computation. In Proceeding of the 23th STOC, pages 410-418, 1991. Google Scholar
  11. J. Radon. Lineare scharen orthogonalen Matrizen. Abh. Math. Sem. Univ. Hamburg, 1(2-14), 1922. Google Scholar
  12. D. B. Shapiro. Quadratic forms and similarities. Bull. Amer. Math. Soc., 81(6), 1975. Google Scholar
  13. D. B. Shapiro. Compositions of quadratic forms. De Gruyter expositions in mathematics 33, 2000. Google Scholar
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