Nonnegative polynomials and sums of squares (Q2892812)

A central question in real algebraic geometry is whether a nonnegative real polynomial in \(n\) variables can be written in a way that makes its nonnegativity apparent.NEWLINENEWLINEIn the seminal paper of \textit{D. Hilbert} [Math. Ann. 32, 342-350 (1888; JFM 20.0198.02)], he showed that every nonnegative polynomial is a sum of squares of polynomials only in the following cases: univariate polynomials, quadratic polynomials and bivariate polynomials of degree 4. In all other cases there are nonnegative polynomials that are not sums of squares and Hilbert's proof used the fact that polynomials of degree \(d\) satisfy linear relations, known as the Cayley-Bacharach relations, which are not satisfied by polynomials of full degree \(2d\).NEWLINENEWLINEIn this paper, it is shown that all linear inequalities that separate nonnegative polynomials from sums of squares come from the Cayley-Bacharach relations in the cases of three variables of degree 4 and two variables of degree 6. Adding an extra variable the author work with homogeneous polynomials (forms).NEWLINENEWLINELet \(H_{n,d}\) be the vector space of real forms in \(n\) variables of degree \(d\). Nonnegative forms and sums of squares of \(H_{n,2d}\) form full-dimensional closed convex cones in \(H_{n,2d}\) which are called \(P_{n,2d}\) and \(\Sigma_{n,2d}\), respectively.NEWLINENEWLINEThe main result states that given a form \(p \in P_{3,6} \setminus \Sigma_{3,6}\), there exist two real homogeneous cubics \(q_1, q_2 \in H_{3,3}\) intersecting in nine (possibly complex) projective points \(z_1, \ldots, z_9\) and a real linear functional \(\ell: H_{3,6} \to {\mathbb R}\) given by NEWLINE\[NEWLINE \ell (f) = \sum_{i=1}^9 \mu_i f(z_i) NEWLINE\]NEWLINE \noindent for some \(\mu_i \in {\mathbb C}\) such that \(\ell(r) \geq 0\) for all \(r \in \Sigma_{3,6}\) and \(\ell(p) < 0\). Furthermore at most two of the points \(z_i\) are complex.NEWLINENEWLINEA similar theorem is proved for the case of forms \(p \in P_{4,4} \setminus \Sigma_{4,4}\).