Irredundant Triangular Decomposition

TL;DR

This paper introduces a randomized algorithm for computing irredundant triangular decompositions of algebraic sets, enabling intrinsic degree bounds and efficient application of Hensel lifting, even in the presence of embedded components.

Contribution

It presents a novel randomized algorithm for irredundant triangular decomposition that overcomes limitations of existing methods by removing embedded components and providing degree bounds.

Findings

Algorithm successfully computes irredundant decompositions.

Provides intrinsic degree bounds for polynomials in the decomposition.

Analyzes success probability of the randomized approach.

Abstract

Triangular decomposition is a classic, widely used and well-developed way to represent algebraic varieties with many applications. In particular, there exist sharp degree bounds for a single triangular set in terms of intrinsic data of the variety it represents, and powerful randomized algorithms for computing triangular decompositions using Hensel lifting in the zero-dimensional case and for irreducible varieties. However, in the general case, most of the algorithms computing triangular decompositions produce embedded components, which makes it impossible to directly apply the intrinsic degree bounds. This, in turn, is an obstacle for efficiently applying Hensel lifting due to the higher degrees of the output polynomials and the lower probability of success. In this paper, we give an algorithm to compute an irredundant triangular decomposition of an arbitrary algebraic set $W$ defined by a set of polynomials in C[x_1, x_2, ..., x_n]. Using this irredundant triangular decomposition, we were able to give intrinsic degree bounds for the polynomials appearing in the triangular sets and apply Hensel lifting techniques. Our decomposition algorithm is randomized, and we analyze the probability of success.