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An Asymptotic Bound on the Composition Number of Integer Sums of Squares Formulas

Published:2011-07-08
Printed: Mar 2013
• P. Hrubeš,
School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540, USA
• A. Wigderson,
School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540, USA
• A. Yehudayoff,
School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540, USA
 Format: LaTeX MathJax PDF

Abstract

Let $\sigma_{\mathbb Z}(k)$ be the smallest $n$ such that there exists an identity $(x_1^2 + x_2^2 + \cdots + x_k^2) \cdot (y_1^2 + y_2^2 + \cdots + y_k^2) = f_1^2 + f_2^2 + \cdots + f_n^2,$ with $f_1,\dots,f_n$ being polynomials with integer coefficients in the variables $x_1,\dots,x_k$ and $y_1,\dots,y_k$. We prove that $\sigma_{\mathbb Z}(k) \geq \Omega(k^{6/5})$.
 Keywords: composition formulas, sums of squares, Radon-Hurwitz number
 MSC Classifications: 11E25 - Sums of squares and representations by other particular quadratic forms

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