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

Published online by Cambridge University Press:  20 November 2018

P. Hrubeš ,
A. Wigderson  and
A. Yehudayoff
P. Hrubeš
Affiliation:
School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540, USA e-mail: pahrubes@gmail.com e-mail: avi@ias.edu e-mail: amir.yehudayoff@gmail.com
A. Wigderson
Affiliation:
School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540, USA e-mail: pahrubes@gmail.com e-mail: avi@ias.edu e-mail: amir.yehudayoff@gmail.com
A. Yehudayoff
Affiliation:
School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540, USA e-mail: pahrubes@gmail.com e-mail: avi@ias.edu e-mail: amir.yehudayoff@gmail.com
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Abstract

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Let ${{\sigma }_{\mathbb{Z}}}\left( k \right)$ be the smallest $n$ such that there exists an identity

$$\left( x_{1}^{2}\,+\,x_{2}^{2}\,+\,\cdot \cdot \cdot \,+\,x_{k}^{2} \right)\,\cdot \,\left( y_{1}^{2}\,+\,y_{2}^{2}\,+\,\cdot \cdot \cdot \,+\,y_{k}^{2} \right)\,=\,f_{1}^{2}\,+\,f_{2}^{2}\,+\,\cdot \cdot \cdot \,+\,f_{n}^{2},$$
with ${{f}_{1}},...,\,{{f}_{n}}$ being polynomials with integer coefficients in the variables ${{x}_{1}},...,\,{{x}_{k}}$ and ${{y}_{1}},...,\,{{y}_{k}}$. We prove that ${{\sigma }_{\mathbb{Z}}}\left( k \right)\,\ge \,\Omega \left( {{k}^{{6}/{5}\;}} \right)$.

Keywords

11E25composition formulassums of squaresRadon-Hurwitz number
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 56 , Issue 1 , 01 March 2013 , pp. 70 - 79
Copyright
Copyright © Canadian Mathematical Society 2013

References

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