Order of contact and ruled submanifolds (Q2043303)

The famous Monge-Cayley-Salmon theorem states that an analytic surface in \(\mathbb{R}^3\) with a dense set of flecnodes (points with a tangent of contact order three) is ruled. It played a crucial role in the solution of the Erdős distinct distances problem by \textit{L. Guth} and \textit{N. H. Katz} [Ann. Math. (2) 181, No. 1, 155--190 (2015; Zbl 1310.52019)]. In this article, the author proves a generalization of the Monge-Cayley-Salmon theorem for an \(m\)-dimensional analytic submanifold \(M \subset \mathbb{R}^n\) admitting a smooth family \(\Gamma_x\), \(x \in M\) of curves with polynomial parametric equations of degree at most \(k\). If for all \(x \in M\) the order of contact between \(\Gamma_x\) and \(M\) at \(x\) is \(k(m+1)\), then \(\Gamma_x \subset M\). The original Monge-Cayley-Salmon theorem is obtained for \(n = 3\), \(m = 2\), and \(k = 1\). The proof considers the volume swept as \(M\) moves along \(\Gamma_x\). On the one hand, this volume is a polynomial of bounded degree while on the other hand the order of contact condition implies a locally slow volume growth. Both conditions together imply vanishing volume whence indeed \(\Gamma_x \subset M\).