# Quadrature for quadrics

## TL;DR

This paper systematically investigates quadrature properties for quadrics, extending traditional methods by considering multi-sheeted branched covers and spherical measures, and clarifies conditions for domains with quadrature properties.

## Contribution

It extends the understanding of quadrature domains for quadrics by incorporating branched coverings and spherical measures, and clarifies a key theorem relating meromorphic extensions to quadrature properties.

## Key findings

- Clarified a theorem linking branched coverings to quadrature domains.
- Analyzed quadrature properties for domains bounded by conic sections.
- Provided new results on hyperbola-based quadrature domains.

## Abstract

We make a systematic investigation of quadrature properties for quadrics, namely integration of holomorphic functions over planar domains bounded by second degree curves. A full understanding requires extending traditional settings by allowing domains which are multi-sheeted, in other words domains which really are branched covering surfaces of the Riemann sphere, and in addition usage of the spherical area measure instead of the Euclidean. The first part of the paper discusses two different points of view of real algebraic curves: traditionally they live in the real projective plane, which is non-orientable, but for their role for quadrature they have to be pushed to the Riemann sphere. The main results include clarifying a previous theorem (joint work with V.~Tkachev), which says that a branched covering map produces a domain with the required quadrature properties if and only it extends to be meromorphic on the double of the parametrizing Riemann surface. In the second half of the paper domains bounded by ellipses, hyperbolas, parabolas and their inverses are studied in detail, with emphasis on the hyperbola case, for which some of the results appear to be new.