Implicitization of Hypersurfaces

TL;DR

This paper introduces new practical algorithms for hypersurface implicitization, efficiently converting parametric descriptions into implicit equations using Gröbner bases, direct computation, and modular methods, with demonstrated experimental performance.

Contribution

The paper presents three novel algorithms for hypersurface implicitization, including a Gröbner basis approach, a direct method, and a modular strategy, enhancing computational efficiency.

Findings

Algorithms are practically efficient, as shown by experimental timings.

The methods handle polynomial and rational parametrizations effectively.

Modular approach reduces high-cost rational arithmetic.

Abstract

We present new, practical algorithms for the hypersurface implicitization problem: namely, given a parametric description (in terms of polynomials or rational functions) of the hypersurface, find its implicit equation. Two of them are for polynomial parametrizations: one algorithm, "ElimTH", has as main step the computation of an elimination ideal via a \textit{truncated, homogeneous} Gr\"obner basis. The other algorithm, "Direct", computes the implicitization directly using an approach inspired by the generalized Buchberger-M\"oller algorithm. Either may be used inside the third algorithm, "RatPar", to deal with parametrizations by rational functions. Finally we show how these algorithms can be used in a modular approach, algorithm "ModImplicit", for avoiding the high costs of arithmetic with rational numbers. We exhibit experimental timings to show the practical efficiency of our new algorithms.