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

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<command> | <command> | ||

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

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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 using this function. The default coefficient field is <tt>Q</tt>, and the default ordering is the length-lexicographic ordering ( | + | 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>ApCoCoA-1:NCo.SetFp|NCo.SetFp</ref>, <ref>ApCoCoA-1:NCo.SetX|NCo.SetX</ref> and <ref>ApCoCoA-1:NCo.SetOrdering|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 ("LLEX"). 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 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, | + | <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,"xy"], [-1, "y"], [1,""]]. The zero polynomial <tt>0</tt> is represented as the empty LIST [].</item> |

<item>@return: a LIST of interreduced polynomials with respect to the current word ordering.</item> | <item>@return: a LIST of interreduced polynomials with respect to the current word ordering.</item> | ||

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

<example> | <example> | ||

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

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

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

NCo.Interreduction(G); | NCo.Interreduction(G); | ||

− | [[[1, | + | [[[1, "a"]], [[1, "b"], [1, ""]], [[1, "c"]]] |

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

</example> | </example> | ||

</description> | </description> | ||

<seealso> | <seealso> | ||

− | <see>NCo.LW</see> | + | <see>ApCoCoA-1:NCo.LW|NCo.LW</see> |

− | <see>NCo.SetFp</see> | + | <see>ApCoCoA-1:NCo.SetFp|NCo.SetFp</see> |

− | <see>NCo.SetOrdering</see> | + | <see>ApCoCoA-1:NCo.SetOrdering|NCo.SetOrdering</see> |

− | <see>NCo.SetX</see> | + | <see>ApCoCoA-1:NCo.SetX|NCo.SetX</see> |

− | <see>Introduction to CoCoAServer</see> | + | <see>ApCoCoA-1:Introduction to CoCoAServer|Introduction to CoCoAServer</see> |

</seealso> | </seealso> | ||

<types> | <types> | ||

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<key>NCo.Interreduction</key> | <key>NCo.Interreduction</key> | ||

<key>Interreduction</key> | <key>Interreduction</key> | ||

− | <wiki-category>Package_gbmr</wiki-category> | + | <wiki-category>ApCoCoA-1:Package_gbmr</wiki-category> |

</command> | </command> |

## Latest revision as of 13:39, 29 October 2020

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

## NCo.Interreduction

Interreduce a LIST of polynomials in a free monoid ring.

### Syntax

NCo.Interreduction(G:LIST):LIST

### Description

Note that, given a 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`.

*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 with respect to the current word ordering.

#### Example

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

### See also