On the real zeros of positive semi-definite biquadratic forms

Ronan Quarez (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes

For a positive semi-definite biquadratic forms $F$ in $(3, 3)$ variables, we prove that if $F$ has a finite number but at least $7$ real zeros $\Z(F)$, then it is not a sum of squares. We show also that if $F$ has at least $11$ zeros, then it has infinitely many real zeros and it is a sum of squares. It can be seen as the counterpart for biquadratic forms as the results of Choi, Lam and Resnick for positive semi-definite ternary sextics.\par We introduce and compute some of the numbers $\BB_{n,m}$ which are set to be equal to $\sup |\Z(F)|$ where $F$ ranges over all the positive semi-definite biquadratic forms $F$ in $(n, m)$ variables such that $|\Z(F)|<\infty$.\par We also recall some old constructions of positive semi-definite biquadratic forms which are not sums of squares and we give some new families of examples.

Mots clés: Biquadratic Forms; Positive semi-definite; Real zeros; Sums of squares

Domaines: Géométrie algébrique [math.AG]

Ronan Quarez. On the real zeros of positive semi-definite biquadratic forms. Communications in Algebra, 2015, 43 (3), pp.1317-1353. ⟨10.1080/00927872.2013.865052⟩. ⟨hal-00523238v2⟩