Two remarks on sums of squares with rational coefficients

Abstract

There exist homogeneous polynomials f with Q-coefficients that are sums of squares over R but not over Q. The only systematic construction of such polynomials that is known so far uses as its key ingredient totally imaginary number fields K/Q with specific Galois-theoretic properties. We first show that one may relax these properties considerably without losing the conclusion, and that this relaxation is sharp at least in a weak sense. In the second part we discuss the open question whether any f as above necessarily has a (non-trivial) real zero. In the minimal open cases (3,6) and (4,4), we prove that all examples without a real zero are contained in a thin subset of the boundary of the sum of squares cone.