Implicitization of Bihomogeneous Parametrizations of Algebraic Surfaces via Linear Syzygies

TL;DR

This paper presents a method to compute the implicit equation of certain algebraic surfaces using linear syzygies, under specific conditions on base points, simplifying the implicitization process.

Contribution

It introduces a novel approach that leverages linear syzygies for implicitization of bi-homogeneous parametrized surfaces with isolated complete intersection base points.

Findings

Implicit equations can be derived from linear syzygies for the specified surfaces.

The method applies when base points are isolated and form a locally complete intersection.

The approach simplifies the implicitization process for these algebraic surfaces.

Abstract

We show that the implicit equation of a surface in 3-dimensional projective space parametrized by bi-homogeneous polynomials of bi-degree (d,d), for a given positive integer d, can be represented and computed from the linear syzygies of its parametrization if the base points are isolated and form locally a complete intersection.