SUMS OF SQUARES OVER TOTALLY REAL FIELDS ARE RATIONAL SUMS OF SQUARES

摘要

Let K be a totally real number field with Galois closure L. We prove that if f is an element of Q[x(1), . . . , x(n)] is a sum of m squares in K[x(1), . . . , x(n)], then f is a sum of 4m . 2([L:Q]+1) (([L:Q])(2) (+ 1)) squares in Q[x(1), . . . , x(n)]. Moreover, our argument is constructive and generalizes to the case of commutative K-algebras. This result gives a partial resolution to a question of Sturmfels on the algebraic degree of certain semidefinite programming problems.