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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. HrubeŇ°,
    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
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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 composition formulas, sums of squares, Radon-Hurwitz number
MSC Classifications: 11E25 show english descriptions Sums of squares and representations by other particular quadratic forms 11E25 - Sums of squares and representations by other particular quadratic forms
 

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