Nonnegative polynomials from vector bundles on real curves (Q2854229)

In the paper under review, ample rank 2 vector bundles on real projective curves (without real points) are used to establish nonnegative polynomials which cannot be written as a sum of squares:NEWLINENEWLINELet \(E\) be a globally generated nontrivial vector bundle of rank 2 on a smooth connected projective curve \(C\) over \(\mathbb{C}\). The cone of sections of \(E\) that vanish somewhere on \(C\) is the zero set of an irreducible homogeneous polynomial \(R_E\) which is called, after [\textit{I. M. Gelfand} et al., Discriminants, resultants, and multidimensional determinants. Boston, MA: Birkhäuser (1994; Zbl 0827.14036)], the \(E\)-resultant. If in addition \(C\) is a real curve (i.e. \(C\) is equipped with an antiholomorphic involution \(\sigma\)) and that \(E\) is a \(\sigma\)-bundle (i.e. there is an antilinear bundle isomorphism \(\tau:E\to E\) that covers \(\sigma\) and satisfies \(\tau^2=1\)) then the restriction of \(R_E\) to the vector space of \(\tau\)-invariant sections gives (after multiplying by a constant) a real polynomial \(\rho_E\) on that vector space. By choosing a suitable multiple, it is assumed that \(\rho_E\) takes at least one positive value. The following is the main result:NEWLINENEWLINEAssume that \(C\) has genus \(g\), that \(E\) is very ample of degree \(d\) and that \(\sigma\) has no fixed points. Then \(\rho_E\) is a nonnegative homogeneous polynomial of degree \(d\), and, if \(d(d-6)\geq 4(g-1)\) and \(E\) has a section that vanishes at exactly two points, then \(\rho_E\) cannot be written as a sums of squares.NEWLINENEWLINEAn explicit example is presented.