# Introduction

The CoCoAIdeal class represents an ideal of polynomials over a given polynomial ring (See <a href="MatlabCoCoAPoly.html">Poly class</a>).

# Constructor

The constructor for a CoCoAIdeal can have two arguments:

• Generators: Array of polynomials
• Optional: 1/0 : 1 = Compute Groebner-Basis of Ideal

If no input is given the default ideal (1) over Q[x] is returned Generators should be a 1xn CoCoAPoly array. A nx1 array is transposed.

# Attributes

 basering A CoCoARing gens an array of polynomials gBasis the Groebner-Basis of an Ideal numBasisElements The number of basis elements. If gBasis is not yet computed the number of generators is stored.

# Methods

 Method Operator Description computeGBasis computes the GBasis of the given Ideal and returns a new ideal with the same set of generators and the computed GBasis. computeLT computes the leading term of the ideal. Returns a CoCoAPoly. display Is used by Matlab to print a CoCoAIdeal to the command window. E.g. when a command is not terminated by a ';'. eq == Compares two ideals. Returns 1 if all generators are equal. ge >= Compares two ideals according to their leading term. Returns 1 if first ideal is greater equal the second ideal. get Returns the attributes of an ideal. Syntax: get(idealVar, 'Keyword'), where idealVar is a CoCoAIdeal and 'Keyword' is one of the keywords: 'BaseRing': Returns the ring over which the polynomials in the generator list are defined. 'Gens': returns the generator array of CoCoAPoly's 'GBasis': returns the GBasis of the Ideal if it has been computed. Otherwise [] is returned. 'NumElements': Returns the number of basis elements. If gBasis is not yet computed the number of generators is returned. 'String': Returns the polynomial as a string. 'Latex' will return a string using Latex syntax. gt > Compares two ideals according to their leading term. Returns 1 if first ideal is greater than the second ideal. isContained Returns 1 if the first ideal is contained in the second ideal. isElem Returns 1 if the polynomial (2. argument) is in the ideal (1. argument) le <= Compares two ideals according to their leading term. Returns 1 if first ideal is less equal the second ideal. lt < Compares two ideals according to their leading term. Returns 1 if first ideal is less than the second ideal. ne ~= returns NOT eq.

# Examples

A collection of examples is provided in the file TestCoCoAIdeal.m. This file defines the function TestCoCoAIdeal which will run and display many examples. A few common examples are presented below.

• Ideal generated by two polynomials.

<matlab> p1 = CoCoAPoly('x^2-1'); p2 = CoCoAPoly('y+x'); myIdeal = CoCoAIdeal([p1 p2]); </matlab>