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.

Gram spectrahedra of ternary quartics

“…, f 5 a basis of span(q) ⊥ . It now follows from [3,Lemma 4.4] that this representation of f is defined over Q. Thus ϑ = 5 i=1 f i ⊗ f i is also rational.…”

“…Remark 6.4. For i = 1, 2, 3, we have R i , R i = 3 4 and for i = 4, 5, 6 we have R i , R i = 1. We show the following theorem concerning the boundary structure of Σ µ K. Theorem 6.5.…”

Facial reduction for exact polynomial sum of squares decomposition

“…We will now prove that the polynomial f does not allow a rational decomposition. The strategy used here is different from the original proof by the authors of [4] and we will use this new proof to extend the example to our main problem. Proposition 3.1.…”

“…In this case, it is not easy to deduce exact values for the two remaining unknowns from the approximations given by SEDUMI. Computing the rank of the principal submatrices of Q, we observe that Q[2,3]× [2,3] has rank 1 and Q[5,6,7]× [5,6,7] has rank 2. Hence we force the determinants of the submatrices Q(t) [2,3]× [2,3] and Q(t) [5,6,7]× [5,6,7] be 0.…”

“…The polynomials obtained in his construction always have common real roots, and the existence of these common roots is central to prove that there is no rational decomposition. A different class of examples have been constructed in [9] and [4]. The strategy used there also produces polynomials with real roots.…”