Ideals of curves given by points

TL;DR

This paper presents a polynomial-time method for computing generators of the ideal defining an algebraic curve from a sufficient number of points, independent of term orderings, with implications for numerical stability.

Contribution

It introduces a term-ordering independent approach to compute ideal generators of algebraic curves efficiently from points.

Findings

Generators can be computed in polynomial time.

Method is independent of term ordering, enhancing numerical stability.

Provides bounds relating generator degrees to curve degree and genus.

Abstract

Let C be an irreducible projective curve of degree d in Pn(K), where K is an algebraically closed field, and let I be the associated homogeneous prime ideal. We wish to compute generators for I, assuming we are given sufficiently many points on the curve C. In particular if I can be generated by polynomials of degree at most m and we are given md + 1 points on C, then we can find a set of generators for I. We will show that a minimal set of generators of I can be constructed in polynomial time. Our constructions are completely independent of any notion of term ordering; this allows us the maximal freedom in performing our constructions in order to improve the numerical stability. We also summarize some classical results on bounds for the degrees of the generators of our ideal in terms of the degree and genus of the curve.