Two remarks on sums of squares with rational coefficients

Volume 121 / 2020

Jose Capco, Claus Scheiderer Banach Center Publications 121 (2020), 25-36 MSC: Primary 14G05; Secondary 11E25, 11R32, 14P99, 90C22. DOI: 10.4064/bc121-2

Abstract

There exist homogeneous polynomials $f$ with $\mathbb Q$-coefficients that are sums of squares over $\mathbb R$ but not over $\mathbb Q$. The only systematic construction of such polynomials that is known so far uses as its key ingredient totally imaginary number fields $K/\mathbb Q$ 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.

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