Quadrature rules with few nodes supported on algebraic curves

Technical Summary: Quadrature Rules with Few Nodes Supported on Algebraic Curves

Problem Statement
The paper addresses the problem of determining the minimal number of nodes required for a quadrature rule of a given strength $s$ to approximate integrals with respect to a positive Borel measure $\mu$ supported on a real algebraic or rational curve. While classical results (Richter, Tchakaloff) guarantee the existence of quadrature rules with at most $\binom{n+s}{s}$ nodes via Carathéodory's Theorem, these bounds are often far from sharp when the support lies on a low-dimensional algebraic set. Specifically, the authors focus on the odd-degree case ($s = 2s' - 1$), which is central to Gaussian quadrature but where the moment cone approach alone fails to yield optimal node counts. The goal is to derive explicit, improved upper bounds on the number of nodes $N$ that depend on the degree of the curve, its topology (specifically places at infinity), and its parametrization.

Methodology
The authors adopt an optimization viewpoint combined with tools from real algebraic geometry.

1. Optimization Framework: Instead of relying solely on convex geometry of moment cones, the authors formulate the existence of a quadrature rule as a minimization problem. They minimize a suitable penalty function (e.g., the sum of weights or a function penalizing node clustering) over the space of quadrature rules satisfying moment matching conditions.
2. Lagrange Multipliers and Stationarity: By analyzing the stationary points of this optimization problem using Lagrange multipliers, the authors derive conditions that the optimal nodes must satisfy. Specifically, optimal nodes correspond to intersection points of the support curve $F=0$ and a derived polynomial curve $G=0$ (or their derivatives), where the curves have parallel tangents.
3. Geometric Counting:
   • For plane curves, the authors apply Bézout's Theorem to bound the number of intersection points between the defining polynomial of the curve and the optimality condition polynomial. They account for multiplicities (nodes appearing with multiplicity $\ge 2$) and contributions from the boundary (places at infinity).
   • For rational curves in arbitrary dimensions, the authors utilize the parametrization $\phi(t)$. They analyze the surjectivity of the pullback map from polynomials on the ambient space to polynomials in the parameter $t$. This allows them to reduce the multivariate problem to a univariate one, leveraging the known optimality of Gaussian quadrature for non-degenerate univariate measures.
4. Singularities and Zeros: The analysis explicitly handles the rational parametrization's denominator $\phi_0(t)$, counting its distinct real zeros (which correspond to places at infinity or singularities) to refine the node bounds.

Key Contributions and Results
The paper provides explicit upper bounds for the minimal number of nodes $N$ for quadrature rules of strength $2s-1$ (and $2s$) supported on curves.

1. Smooth Plane Algebraic Curves:
   For a smooth, irreducible plane curve $C \subset \mathbb{R}^2$ of degree $d$ with at most $t$ distinct places at infinity, the authors prove the existence of a quadrature rule of strength $2s-1$ with at most:
   $$N \le ds - \left\lfloor \frac{d}{2} \right\rfloor + 1 + 2t$$
   If the curve is compact ($t=0$), this simplifies to $N \le ds - \lceil d/2 \rceil + 1$. This improves upon the previous bound of $N \le ds$ established by Riener and Schweighofer.

2. Rational Curves in Arbitrary Dimensions:
   For a rational curve $C \subset \mathbb{R}^n$ of degree $d$ with a polynomial parametrization $\phi(t) = (\phi_1/\phi_0, \dots, \phi_n/\phi_0)$ where $\max \deg \phi_i = D$ and $\phi_0$ has $p$ distinct real zeros:

   • Strength $2s-1$: There exists a rule with at most $N \le Ds - \lfloor D/2 \rfloor + p + 1$ nodes.
   • Strength $2s$: There exists a rule with at most $N \le Ds + p + 1$ nodes.
     These results recover the classical Gaussian bound ($N=s$) for lines ($D=1$) and generalize results by Zalar for specific plane curves to arbitrary rational curves in higher dimensions.

3. Specific Curve Families:
   The paper analyzes specific cases, such as the curve $y = x^d$ with non-negative nodes. For this case, they derive a bound of $N \le \lfloor \frac{ds-1}{2} - \frac{d(d-3)}{4} \rfloor + 1$, which is shown to be asymptotically sharper than existing bounds for large $d$.

Significance and Claims
The authors claim that their work establishes a direct link between the algebraic complexity of the supporting curve and the minimal size of quadrature formulas.

• Unification: The results provide a unified framework connecting real algebraic geometry, polynomial optimization, and moment theory.
• Improvement: The derived bounds strictly improve upon classical Carathéodory-type bounds and previous results by Riener–Schweighofer and Zalar, particularly by incorporating geometric features like the number of places at infinity and the specific nature of rational parametrizations.
• Odd-Degree Focus: The paper highlights that the odd-degree case ($2s-1$) requires distinct geometric and optimization arguments compared to the even-degree case, as the moment cone does not fully capture the minimal node count in this setting.
• Optimality: The authors note that for generic rational curves in dimensions $n \ge 3$, their bounds are optimal, matching the lower bounds derived from the univariate Gaussian case.

The paper concludes by acknowledging that while the bounds are tight in specific cases (lines, certain low-degree curves), their optimality for all degrees and strengths remains an open question, suggesting future work involving finer invariants like singularity structure and genus.