# Difference between revisions of "ApCoCoA-1:NCo.PrefixNR"

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<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them. | <em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them. | ||

<par/> | <par/> | ||

− | Please set ring environment coefficient field <tt> K</tt>, alphabet (or set of indeterminates) <tt>X</tt>, rewrite relations <tt>Relations</tt> and word ordering <tt>Ordering</tt> through the functions <ref>ApCoCoA-1:NCo.SetFp|NCo.SetFp</ref>, <ref>ApCoCoA-1:NCo.SetX|NCo.SetX</ref>, <ref>ApCoCoA-1:NCo.SetRelations|NCo.SetRelations</ref> and <ref>ApCoCoA-1:NCo.SetOrdering|NCo.SetOrdering</ref>, respectively, before using this function. The default coefficient field is the field of rational numbers, i.e. RAT in CoCoAL, and the default ordering is the length-lexicographic ordering | + | Please set ring environment coefficient field <tt> K</tt>, alphabet (or set of indeterminates) <tt>X</tt>, rewrite relations <tt>Relations</tt> and word ordering <tt>Ordering</tt> through the functions <ref>ApCoCoA-1:NCo.SetFp|NCo.SetFp</ref>, <ref>ApCoCoA-1:NCo.SetX|NCo.SetX</ref>, <ref>ApCoCoA-1:NCo.SetRelations|NCo.SetRelations</ref> and <ref>ApCoCoA-1:NCo.SetOrdering|NCo.SetOrdering</ref>, respectively, before using this function. The default coefficient field is the field of rational numbers, i.e. RAT in CoCoAL, and the default ordering is the length-lexicographic ordering "LLEX". For more information, please check the relevant functions. |

<itemize> | <itemize> | ||

− | <item>@param <em>F:</em> a LIST which represents a polynomial in the monoid ring. Each polynomial is represented as a LIST of LISTs, i.e. as <tt>[[C1,W1],...,[Cs,Ws]]</tt> where, for each i, Wi is a term represented as a STRING and Ci is the coefficient of Wi. For example, polynomial <tt>f=xy-y+1</tt> is represented as F:=[[1, | + | <item>@param <em>F:</em> a LIST which represents a polynomial in the monoid ring. Each polynomial is represented as a LIST of LISTs, i.e. as <tt>[[C1,W1],...,[Cs,Ws]]</tt> where, for each i, Wi is a term represented as a STRING and Ci is the coefficient of Wi. For example, polynomial <tt>f=xy-y+1</tt> is represented as F:=[[1,"xy"], [-1, "y"], [1,""]]. The zero polynomial <tt>0</tt> is represented as the empty LIST [].</item> |

<item>@param <em>G</em>: a LIST of polynomials.</item> | <item>@param <em>G</em>: a LIST of polynomials.</item> | ||

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</itemize> | </itemize> | ||

<example> | <example> | ||

− | NCo.SetX( | + | NCo.SetX("abc"); |

− | NCo.SetOrdering( | + | NCo.SetOrdering("LLEX"); |

− | NCo.SetRelations([[ | + | NCo.SetRelations([["aa",""], ["bb",""], ["ab","c"], ["ac", "b"], ["cb", "a"]]); |

− | F := [[1, | + | F := [[1,"ba"]]; |

− | G:=[[[1, | + | G:=[[[1,"b"],[1,""]], [[1,"c"]]]; |

NCo.PrefixNR(F, G); | NCo.PrefixNR(F, G); | ||

− | [[1, | + | [[1, "a"]] |

------------------------------- | ------------------------------- | ||

</example> | </example> |

## Latest revision as of 13:44, 29 October 2020

This article is about a function from ApCoCoA-1. |

## NCo.PrefixNR

The prefix normal remainder of a polynomial with respect to a LIST of polynomials in a finitely presented monoid ring.

### Syntax

NCo.PrefixNR(F:LIST, G:LIST):LIST

### Description

*Please note:* The function(s) explained on this page is/are using the *ApCoCoAServer*. You will have to start the ApCoCoAServer in order to use it/them.

Please set ring environment coefficient field ` K`, alphabet (or set of indeterminates) `X`, rewrite relations `Relations` and word ordering `Ordering` through the functions NCo.SetFp, NCo.SetX, NCo.SetRelations and NCo.SetOrdering, respectively, before using this function. The default coefficient field is the field of rational numbers, i.e. RAT in CoCoAL, and the default ordering is the length-lexicographic ordering "LLEX". For more information, please check the relevant functions.

@param

*F:*a LIST which represents a polynomial in the monoid ring. Each polynomial is represented as a LIST of LISTs, i.e. as`[[C1,W1],...,[Cs,Ws]]`where, for each i, Wi is a term represented as a STRING and Ci is the coefficient of Wi. For example, polynomial`f=xy-y+1`is represented as F:=[[1,"xy"], [-1, "y"], [1,""]]. The zero polynomial`0`is represented as the empty LIST [].@param

*G*: a LIST of polynomials.@return: a LIST which represents the prefix normal remainder of F with respect to G.

#### Example

NCo.SetX("abc"); NCo.SetOrdering("LLEX"); NCo.SetRelations([["aa",""], ["bb",""], ["ab","c"], ["ac", "b"], ["cb", "a"]]); F := [[1,"ba"]]; G:=[[[1,"b"],[1,""]], [[1,"c"]]]; NCo.PrefixNR(F, G); [[1, "a"]] -------------------------------

### See also