Generalizations of the odd degree theorem and applications (Q1424117)

The authors present sufficient conditions to guarantee that a linear subspace \(L\subset {\mathbb P}{\mathbb R}^n\) has a non empty intersection with an algebraic variety \(V\subset{\mathbb P}{\mathbb R}^n\) whose complexification \(V_{\mathbb C}\subset {\mathbb P}^n\) is irreducible of codimension \(m\geq 1\). This allows them to show that any six dimensional subspace of \(5\times 5\) real symmetric matrices contains a nonzero matrix of rank at most \(3\). This numerical evidence supports a conjecture, proposed by \textit{D. Falikman, S. Friedland} and \textit{R. Loewy} [Pac. J. Math. 207, No. 1, 157--176 (2002; Zbl 1050.15018)], which says that the minimal dimension for which a subspace of \(5\times 5\) real symmetric matrices contains a nonzero matrix of rank at most \(3\) is \(6\). The conditions mentioned above involve the introduction of a generalization to the even degree case of the odd degree theorem, which says that: If \(V\subset {\mathbb P}{\mathbb R}^n\) is an algebraic variety whose complexification \(V_{\mathbb C}\) is irreducible of codimension \(m>1\) and odd degree, then any linear subspace \(L\subset {\mathbb P}{\mathbb R}^n\) of dimension \(m\) intersects \(V\) in a non empty set.