# 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', 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. Available are:
• Q: Ring of rational numbers
• Z: Ring of integers
• RF:
• RD:
• 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 degree compatible term ordering.

# Methods

The following methods are available for rings:

• 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. This function is called when using the '==' operator.
• neq: returns NOT eq. This function is called when using the '~=' operator
• 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>