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

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

<itemize> | <itemize> | ||

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

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<see>NC.Intersection</see> | <see>NC.Intersection</see> | ||

<see>NC.IsGB</see> | <see>NC.IsGB</see> | ||

+ | <see>NC.IsHomog</see> | ||

<see>NC.KernelOfHomomorphism</see> | <see>NC.KernelOfHomomorphism</see> | ||

<see>NC.LC</see> | <see>NC.LC</see> |

## Revision as of 17:38, 7 June 2012

## NC.NR

Normal remainder polynomial with respect to a list of polynomials over a free monoid ring.

### Syntax

NC.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 NC.SetFp, NC.SetX and NC.SetOrdering, respectively, before calling the function. The default coefficient field is `Q`. The default ordering is 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 pairs 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 an empty LIST [].@param

*G*: a LIST of non-zero polynomials in`K<X>`.@return: a LIST which represents the normal remainder of

`F`w.r.t.`G`.

#### Example

NC.SetX(<quotes>abc</quotes>); NC.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]; NC.NR(F,G); [[1, <quotes>bcb</quotes>], [-1, <quotes>ccc</quotes>], [-1, <quotes>bb</quotes>], [1, <quotes>cb</quotes>], [-1, <quotes></quotes>]] ------------------------------- NC.SetOrdering(<quotes>ELIM</quotes>); NC.NR(F,G); [[1, <quotes>bcb</quotes>], [-1, <quotes>bb</quotes>], [1, <quotes>cb</quotes>], [-1, <quotes>ccc</quotes>], [-1, <quotes></quotes>]] -------------------------------

### See also