We investigate quadrature rules for measures supported on real algebraic and rational curves, focusing on the odd-degree case 2 s − 1 . Adopting an optimization viewpoint, we minimize suitable penalty functions over the space of quadrature rules of strength 2 s − 1 , so that optimal solutions yield rules with the minimal number of nodes. For plane algebraic curves of degree d , we derive explicit node bounds depending on d and the number of places at infinity, improving results of Riener–Schweighofer, and Zalar. For rational curves in arbitrary dimension of degree d , we further refine these bounds using the geometry of the parametrization and recover the classical Gaussian quadrature bound when d = 1 . Our results reveal a direct link between the algebraic complexity of the supporting curve and the minimal size of quadrature formulas, providing a unified framework that connects real algebraic geometry, polynomial optimization, and moment theory.

A singular k-tuple of a tensor T of format (n1,…,nk) is essentially a complex critical point of the distance function from T constrained to the cone of tensors of format (n1,…,nk) of rank at most one. A generic tensor has finitely many complex singular k-tuples, and their number depends only on the tensor format. Furthermore, if we fix the first k−1 dimensions ni, then the number of singular k-tuples of a generic tensor becomes a monotone non-decreasing function in one integer variable nk, that stabilizes when (n1,…,nk) reaches a boundary format. In this paper, we study the linear span of singular k-tuples of a generic tensor. Its dimension also depends only on the tensor format. In particular, we concentrate on special order three tensors and order-k tensors of format (2,…,2,n). As a consequence, if again we fix the first k−1 dimensions ni and let nk increase, we show that in these special formats, the dimension of the linear span stabilizes as well, but at some concise non-sub-boundary format. We conjecture that this phenomenon holds for an arbitrary format with k>3. Finally, we provide equations for the linear span of singular triples of a generic order three tensor T of some special non-sub-boundary format. From these equations, we conclude that T belongs to the linear span of its singular triples, and we conjecture that this is the case for every tensor format.