Geodesic
An Asymptotic Bound on the Composition Number of Integer Sums of Squares Formulas
Canadian mathematical bulletin, Tome 56 (2013) no. 1, pp. 70-79

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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 ${{\sigma }_{\mathbb{Z}}}\left( k \right)\,\ge \,\Omega \left( {{k}^{{6}/{5}\;}} \right)$ .
DOI : 10.4153/CMB-2011-143-x
Mots-clés : 11E25, composition formulas, sums of squares, Radon-Hurwitz number 
Hrubeš, P.; Wigderson, A.; Yehudayoff, A. An Asymptotic Bound on the Composition Number of Integer Sums of Squares Formulas. Canadian mathematical bulletin, Tome 56 (2013) no. 1, pp. 70-79. doi: 10.4153/CMB-2011-143-x
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