An asymptotic bound on the composition number of integer sums of squares formulas

摘要

Let Z(k) be the smallest n such that there exists an identity (x ~2 + x_2 ~2 + . . . + x _k ~2).(y _1 ~2+ y_2 ~2 + . . . + y _k ~2) = f _1 ~2+ f_2 ~2 + . . . + 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 Z(k) ≥(k~(6/5)).