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 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.