# ApCoCoA:MatlabCoCoARing

# Introduction

The CoCoARing class represents a polynomial ring structure. It stores the name of indeterminates defined over a coefficient ring which is defined in the typ attribute amongst other supporting attributes.

# Constructor

The constructor function takes a parameter in the form 'keyword',value. For example to build a ring over Q use the keyword = 'Typ' and the value = 'Q'. The function is called CoCoARing().

Available keywords are listed below:

Keyword | Possible Values | Default |

Typ | 'Q','Z','RF','RD' | 'Q' |

Order | 0,1,2,... | 0 |

Noindets | 1,2,... | 1 |

Indets | 'auto', {'anyString1','anyString2',...} | {'x[1]'} |

Ordering | 'DegRevLex', 'DegLex', 'Lex', 'RevLex' , AnyTermOrderingMatrix | 1 |

Table 1: Keywords for CoCoARing

Note: The order of the keywords is not completly arbitrary:

- The keyword
*Ordering*can only be used after the keyword*Indets*. *Noindets*should be defined before*Indets*otherwise result might not be correct as default value is used.

# Attributes

typ | Defines the coefficent ring. In the current implementation only Q - Ring of rational numbers - is available. |

order | Defines the order of the polynmoial ring. Usually a prime number or 0. |

noindets | Stores the number of indeterminates. |

indets | Holds the label of the indeterminates as a 1xNoindets cell array of strings. |

ordering | A matrix representing a term ordering. |

# Methods

The following methods are available for rings:

Method | Operator | Description |

display | Is used by Matlab to print a CoCoARing to the command window. E.g. when a command is not terminated by a ';'. | |

eq | == | Compares two rings and returns 1 in case that all attributes of the two rings are equal. Otherwise 0 is returned. |

neq | ~= | returns NOT eq. |

get | Returns the attributes of a ring. Syntax: get(ringVar, 'Keyword'), where ringVar is a CoCoARing and 'Keyword' is one of the keywords listed in table 1. Additionaly the keyword 'String' can be used. This will return the polynomial as a string. | |

one | returns the 1 polynomial of the polynomial ring | |

zero | returns the 0 polynomial of the polynomial ring |

# Examples

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

- Default ring Q[x[1]]:

<matlab> defaultRing = CoCoARing(); </matlab>

- Q[x1,...,xN]: The only supported ring for the CoCoAPoly class at the moment is Q[x1,...,xN]. To generate this ring with a DegRevLex ordering over 3 indeterminates x,y and z use the following code:

<matlab> myR = CoCoARing('Typ','Q','Indets',{'x','y','z'},'Ordering','DegRevLex'); </matlab>

- To generate the 1 and 0 polynomial of a CoCoARing use the following code:

<matlab> oneOfRing = one(myR); zeroOfRing = zero(myR); </matlab>

- To read the attributes of a CoCoARing use the following command (this examples reads all attributes):

<matlab> [a1,a2,a3,a4,a5] = get(myR,'Typ','Order','Noindets','Indets','Ordering'); </matlab>