Quadrature rules with few nodes supported on algebraic curves
Cordian Riener, Ettore Teixeira Turatti
TL;DR
We address constructing quadrature rules for measures supported on real algebraic and rational curves, focusing on the odd-degree case $2s-1$. An optimization framework minimizes a penalty over quadrature rules of strength $2s-1$, linking node minimization to the geometry of the supporting curve. Explicit node bounds are derived: for plane curves of degree $d$ with $t$ places at infinity, $N \le ds - \lceil d/2 \rceil + 1 + 2t$; for rational curves with parametrization degree $D$ and $p$ real zeros of the denominator, $N \le Ds - \lceil D/2 \rceil + p + 1$ for strength $2s-1$ and $N \le Ds + p + 1$ for strength $2s$, with $D=1$ recovering Gaussian bound $N=s$. This unifies real algebraic geometry, polynomial optimization, and moment theory, clarifying how the algebraic complexity of the support governs minimal quadrature sizes and enabling geometry-aware, sharper bounds.
Abstract
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.