Quadrature rules with few nodes supported on algebraic curves

We investigate quadrature rules for measures supported on real algebraic and rational curves, focusing on the {odd-degree} case \(2s-1\). Adopting an optimization viewpoint, we minimize suitable penalty functions over the space of quadrature rules of strength \(2s-1\), 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.