Dualization of projective algebraic sets by using Gr\"obner bases elimination techniques

TL;DR

This paper presents a method for dualizing projective algebraic sets using Gröbner bases for variable elimination, with an implementation in Singular and illustrative examples.

Contribution

It introduces a novel dualization algorithm for projective algebraic sets based on Gröbner bases and demonstrates its implementation and applications.

Findings

Effective dualization of projective algebraic sets demonstrated

Algorithm implemented in Singular with practical examples

Clarifies relationships between ideals, radicals, and dual ideals

Abstract

The set of common roots of a finite set $I$ (it is an ideal) of homogeneous polynomials is known as projective algebraic set $V$. In this article I show how to dualize such projective algebraic sets $V$ by elimination of variables from a system of polynomials with the Gr\"obner bases method. A dualization algorithm is implemented in the computer algebra system {\sc Singular}. Some examples are given. The main diagram shows the relationship between the ideal $I$, its radical $\sqrt{I}$ and their dual ideals.