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.