# Bivariate Semialgebraic Splines

## TL;DR

This paper investigates the dimension of bivariate semialgebraic spline spaces over meshes with algebraic curve edges, providing formulas and bounds for various cases using algebraic methods.

## Contribution

It introduces new dimension formulas for semialgebraic spline spaces in extreme and generic cases, extending prior work to more complex mesh configurations.

## Key findings

- Dimension formulas for specific algebraic curve meshes
- Bounds on degree for formula validity
- Analysis of non-extreme mesh cases

## Abstract

Semialgebraic splines are bivariate splines over meshes whose edges are arcs of algebraic curves. They were first considered by Wang, Chui, and Stiller. We compute the dimension of the space of semialgebraic splines in two extreme cases. If the polynomials defining the edges span a three-dimensional space of polynomials, then we compute the dimensions from the dimensions for a corresponding rectilinear mesh. If the mesh is sufficiently generic, we give a formula for the dimension of the spline space valid in large degree and bound how large the degree must be for the formula to hold. We also study the dimension of the spline space in examples which do not satisfy either extreme. The results are derived using commutative and homological algebra.