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

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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 <em>alphabet</em> (or set of indeterminates) <tt>X</tt> and <em>word ordering</em> via the functions <ref>ApCoCoA-1:NCo.SetX|NCo.SetX</ref> and <ref>ApCoCoA-1:NCo.SetOrdering|NCo.SetOrdering</ref>, respectively, before calling this function. The default ordering is the length-lexicographic ordering ( | + | Please set ring environment <em>alphabet</em> (or set of indeterminates) <tt>X</tt> and <em>word ordering</em> via the functions <ref>ApCoCoA-1:NCo.SetX|NCo.SetX</ref> and <ref>ApCoCoA-1:NCo.SetOrdering|NCo.SetOrdering</ref>, respectively, before calling this function. 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 polynomial in the free monoid ring <tt>F_{2}<X></tt>. Each polynomial is represented as a LIST of words (or terms) in <tt><X></tt>. Each word is represented as a STRING. For example, <tt>xy^2x</tt> is represented as | + | <item>@param <em>F:</em> a polynomial in the free monoid ring <tt>F_{2}<X></tt>. Each polynomial is represented as a LIST of words (or terms) in <tt><X></tt>. Each word is represented as a STRING. For example, <tt>xy^2x</tt> is represented as "xyyx", and the identity is represented as the empty string "". Thus, the polynomial <tt>f=xy-y+1</tt> is represented as F:=["xy", "y", ""]. The zero polynomial <tt>0</tt> is represented as the empty LIST [].</item> |

<item>@param <em>G:</em> a LIST of polynomials in the free monoid ring <tt>F_{2}<X></tt>.</item> | <item>@param <em>G:</em> a LIST of polynomials in the free monoid ring <tt>F_{2}<X></tt>.</item> | ||

<item>@return: a LIST which represents the normal remainder of F with respect to G.</item> | <item>@return: a LIST which represents the normal remainder of F with respect to G.</item> | ||

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

<example> | <example> | ||

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

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

− | F1:=[ | + | F1:=["xX",""]; |

− | F2:=[ | + | F2:=["Xx",""]; |

− | F3:=[ | + | F3:=["yY",""]; |

− | F4:=[ | + | F4:=["Yy",""]; |

G:=[F1,F2,F3,F4]; | G:=[F1,F2,F3,F4]; | ||

− | F:=[ | + | F:=["xyYxxXYY"]; |

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

− | [ | + | ["xxYY"] |

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

</example> | </example> |

## Latest revision as of 13:38, 29 October 2020

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

## NCo.BNR

The normal remainder of a polynomial with respect to a LIST of polynomials in a free monoid ring over the binary field F_{2}={0,1}.

### Syntax

NCo.BNR(F:LIST, G:LIST):INT

### 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 *alphabet* (or set of indeterminates) `X` and *word ordering* via the functions NCo.SetX and NCo.SetOrdering, respectively, before calling this function. The default ordering is the length-lexicographic ordering ("LLEX"). For more information, please check the relevant functions.

@param

*F:*a polynomial in the free monoid ring`F_{2}<X>`. Each polynomial is represented as a LIST of words (or terms) in`<X>`. Each word is represented as a STRING. For example,`xy^2x`is represented as "xyyx", and the identity is represented as the empty string "". Thus, the polynomial`f=xy-y+1`is represented as F:=["xy", "y", ""]. The zero polynomial`0`is represented as the empty LIST [].@param

*G:*a LIST of polynomials in the free monoid ring`F_{2}<X>`.@return: a LIST which represents the normal remainder of F with respect to G.

#### Example

NCo.SetX("xyXY"); NCo.SetOrdering("LLEX"); F1:=["xX",""]; F2:=["Xx",""]; F3:=["yY",""]; F4:=["Yy",""]; G:=[F1,F2,F3,F4]; F:=["xyYxxXYY"]; NCo.BNR(F,G); ["xxYY"] -------------------------------

### See also