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

http://dx.doi.org/10.4153/CMB-2011-143-x
Canad. Math. Bull. 56(2013), 70-79

  Published:2011-07-08
 Printed: Mar 2013

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 $\sigma_{\mathbb Z}(k) \geq \Omega(k^{6/5})$.