Implicitization of rational hypersurfaces via linear syzygies: a practical overview

TL;DR

This paper provides a practical overview of syzygy-based algorithms for implicitizing rational hypersurfaces, translating complex algebraic methods into accessible computational procedures.

Contribution

It simplifies the theoretical framework of syzygy-based implicitization algorithms, making them more accessible for practical implementation.

Findings

Concrete description of syzygy-based algorithms

Connection to the moving curves method

Avoidance of advanced homological algebra tools

Abstract

We unveil in concrete terms the general machinery of the syzygy-based algorithms for the implicitization of rational surfaces in terms of the monomials in the polynomials defining the parametrization, following and expanding our joint article with M. Dohm. These algebraic techniques, based on the theory of approximation complexes due to J. Herzog, A, Simis and W. Vasconcelos, were introduced for the implicitization problem by J.-P. Jouanolou, L. Bus\'e, and M. Chardin. Their work was inspired by the practical method of moving curves, proposed by T. Sederberg and F. Chen, translated into the language of syzygies by D. Cox. Our aim is to express the theoretical results and resulting algorithms into very concrete terms, avoiding the use of the advanced homological commutative algebra tools which are needed for their proofs.