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

(New page: <command> <title>NCo.NR</title> <short_description> Normal remainder of polynomial with respect to a list of polynomials in a free monoid ring. </short_description> <syntax> NCo.NR(F:LIST,...) |
|||

Line 2: | Line 2: | ||

<title>NCo.NR</title> | <title>NCo.NR</title> | ||

<short_description> | <short_description> | ||

− | + | The normal remainder of a polynomial with respect to a LIST of polynomials in a free monoid ring. | |

</short_description> | </short_description> | ||

<syntax> | <syntax> | ||

Line 10: | Line 10: | ||

<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>coefficient field</em> <tt>K</tt>, <em>alphabet</em> (or set of indeterminates) <tt>X</tt> and <em>ordering</em> via the functions <ref>NCo.SetFp</ref>, <ref>NCo.SetX</ref> and <ref>NCo.SetOrdering</ref>, respectively, before | + | Please set ring environment <em>coefficient field</em> <tt> K</tt>, <em>alphabet</em> (or set of indeterminates) <tt>X</tt> and <em>ordering</em> via the functions <ref>NCo.SetFp</ref>, <ref>NCo.SetX</ref> and <ref>NCo.SetOrdering</ref>, respectively, before using this function. The default coefficient field is <tt>Q</tt>, and the default ordering is the length-lexicographic ordering (<quotes>LLEX</quotes>). For more information, please check the relevant functions. |

<itemize> | <itemize> | ||

<item></item> | <item></item> | ||

− | <item>@param <em>F</em>: a polynomial in <tt>K<X></tt>. Each polynomial is represented as a LIST of monomials, which are | + | <item>@param <em>F</em>: a polynomial in <tt>K<X></tt>. Each polynomial is represented as a LIST of monomials, which are LISTs of the form [C, W] where W is a word in <tt><X></tt> and C is the coefficient of W. For example, the polynomial <tt>f=xy-y+1</tt> is represented as F:=[[1,<quotes>xy</quotes>], [-1, <quotes>y</quotes>], [1,<quotes></quotes>]]. The zero polynomial <tt>0</tt> is represented as the empty LIST [].</item> |

<item>@param <em>G</em>: a LIST of non-zero polynomials in <tt>K<X></tt>.</item> | <item>@param <em>G</em>: a LIST of non-zero polynomials in <tt>K<X></tt>.</item> | ||

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

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

<example> | <example> |

## Revision as of 16:46, 29 April 2013

## NCo.NR

The normal remainder of a polynomial with respect to a LIST of polynomials in a free monoid ring.

### Syntax

NCo.NR(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` and *ordering* via the functions NCo.SetFp, NCo.SetX and NCo.SetOrdering, respectively, before using this function. The default coefficient field is `Q`, and the default ordering is the length-lexicographic ordering ("LLEX"). For more information, please check the relevant functions.

@param

*F*: a polynomial in`K<X>`. Each polynomial is represented as a LIST of monomials, which are LISTs of the form [C, W] where W is a word in`<X>`and C is the coefficient of W. For example, the 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 non-zero polynomials in`K<X>`.@return: a LIST which represents the normal remainder of

`F`with respect to`G`.

#### Example

NCo.SetX(<quotes>abc</quotes>); NCo.RingEnv(); Coefficient ring : Q Alphabet : abc Ordering : LLEX ------------------------------- F:=[[1,<quotes>ab</quotes>],[1,<quotes>aca</quotes>],[1,<quotes>bb</quotes>],[1,<quotes>bab</quotes>],[1,<quotes></quotes>]]; F1 := [[1,<quotes>a</quotes>],[1,<quotes>c</quotes>]]; F2 := [[1,<quotes>b</quotes>],[1,<quotes>ba</quotes>]]; G:=[F1,F2]; NCo.NR(F,G); [[1, <quotes>bcb</quotes>], [-1, <quotes>ccc</quotes>], [-1, <quotes>bb</quotes>], [1, <quotes>cb</quotes>], [-1, <quotes></quotes>]] ------------------------------- NCo.SetOrdering(<quotes>ELIM</quotes>); NCo.NR(F,G); [[1, <quotes>bcb</quotes>], [-1, <quotes>bb</quotes>], [1, <quotes>cb</quotes>], [-1, <quotes>ccc</quotes>], [-1, <quotes></quotes>]] -------------------------------

### See also