ApCoCoA-1:MatlabCoCoAIdeal
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[1]] 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>
