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

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<title>NCo.Interreduction</title> | <title>NCo.Interreduction</title> | ||

<short_description> | <short_description> | ||

− | Interreduce a | + | Interreduce a LIST of polynomials in a free monoid ring. |

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

− | Note that, given an | + | Note that, given an word ordering <tt>Ordering</tt>, a set of non-zero polynomials <tt>G</tt> is called <em>interreduced</em> with respect to <tt>Ordering</tt> if no element of <tt>Supp(g)</tt> is contained in the leading word ideal <tt>LW(G\{g})</tt> for all <tt>g</tt> in <tt>G</tt>. |

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

<syntax> | <syntax> | ||

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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>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>@param <em>G</em>: a LIST of polynomials in <tt>K<X></tt>. Each polynomial is represented as a LIST of monomials, which are | + | <item>@param <em>G</em>: a LIST of polynomials 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>@return: a LIST of interreduced polynomials.</item> | <item>@return: a LIST of interreduced polynomials.</item> | ||

</itemize> | </itemize> |

## Revision as of 16:52, 29 April 2013

## NCo.Interreduction

Interreduce a LIST of polynomials in a free monoid ring.

Note that, given an word ordering `Ordering`, a set of non-zero polynomials `G` is called *interreduced* with respect to `Ordering` if no element of `Supp(g)` is contained in the leading word ideal `LW(G\{g})` for all `g` in `G`.

### Syntax

NCo.Interreduction(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

*G*: a LIST of polynomials 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 [].@return: a LIST of interreduced polynomials.

#### Example

NCo.SetX(<quotes>abc</quotes>); NCo.SetOrdering(<quotes>ELIM</quotes>); G:=[[[1,<quotes>ba</quotes>]], [[1,<quotes>b</quotes>],[1,<quotes></quotes>]], [[1,<quotes>c</quotes>]]]; NCo.Interreduction(G); [[[1, <quotes>a</quotes>]], [[1, <quotes>b</quotes>], [1, <quotes></quotes>]], [[1, <quotes>c</quotes>]]] -------------------------------

### See also