# Computing strong regular characteristic pairs with Groebner bases

## TL;DR

This paper introduces a method to decompose polynomial ideals into strong regular characteristic pairs using Groebner bases, enabling effective zero set representations and ideal analysis.

## Contribution

It presents a novel algorithm for decomposing polynomial sets into strong regular characteristic pairs via Groebner basis computations, facilitating ideal analysis and solution representations.

## Key findings

- Algorithm successfully decomposes polynomial sets into strong regular characteristic pairs.
- Provides two types of zero set representations: via Groebner bases and regular triangular sets.
- Experimental results demonstrate the efficiency and effectiveness of the proposed method.

## Abstract

The W-characteristic set of a polynomial ideal is the minimal triangular set contained in the reduced lexicographical Groebner basis of the ideal. A pair (G,C) of polynomial sets is a strong regular characteristic pair if G is a reduced lexicographical Groebner basis, C is the W-characteristic set of the ideal <G>, the saturated ideal sat(C) of C is equal to <G>, and C is regular. In this paper, we show that for any polynomial ideal I with given generators one can either detect that I is unit, or construct a strong regular characteristic pair (G,C) by computing Groebner bases such that I$\subseteq$sat(C)=<G> and sat(C) divides I, so the ideal I can be split into the saturated ideal sat(C) and the quotient ideal I:sat(C). Based on this strategy of splitting by means of quotient and with Groebner basis and ideal computations, we devise a simple algorithm to decompose an arbitrary polynomial set F into finitely many strong regular characteristic pairs, from which two representations for the zeros of F are obtained: one in terms of strong regular Groebner bases and the other in terms of regular triangular sets. We present some properties about strong regular characteristic pairs and characteristic decomposition and illustrate the proposed algorithm and its performance by examples and experimental results.