Semialgebraic Splines

TL;DR

This paper investigates the mathematical properties of bivariate semialgebraic splines, focusing on their dimensionality in specific configurations, and explores their Hilbert functions and polynomials in various cases.

Contribution

It formulates the spaces of semialgebraic splines using graded modules and computes their dimensions in key extreme cases with a single interior vertex.

Findings

Dimension formulas for splines with large degree in extreme cases

Analysis of Hilbert functions and polynomials for specific vertex configurations

Examples illustrating the behavior of splines beyond extreme cases

Abstract

Semialgebraic splines are functions that are piecewise polynomial with respect to a cell decomposition into sets defined by polynomial inequalities. We study bivariate semialgebraic splines, formulating spaces of semialgebraic splines in terms of graded modules. We compute the dimension of the space of splines with large degree in two extreme cases when the cell decomposition has a single interior vertex. First, when the forms defining the edges span a two-dimensional space of forms of degree n---then the curves they define meet in n^2 points in the complex projective plane. In the other extreme, the curves have distinct slopes at the vertex and do not simultaneously vanish at any other point. We also study examples of the Hilbert function and polynomial in cases of a single vertex where the curves do not satisfy either of these extremes.