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

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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 | + | Please set non-commutative polynomial ring (via the command <ref>Use</ref>) and word ordering (via the function <ref>NC.SetOrdering</ref>) before calling this function. The default word ordering is the length-lexicographic ordering (<quotes>LLEX</quotes>). For more information, please check the relevant commands and functions. |

<itemize> | <itemize> | ||

− | <item>@param <em>G</em>: a LIST of polynomials | + | <item>@param <em>G</em>: a LIST of non-commutative polynomials. Each polynomial is represented as a LIST of LISTs, and each element in every inner LIST involves only one indeterminate or none (a constant). For example, the polynomial <tt>f=2x[2]y[1]x[2]^2-9y[2]x[1]^2x[2]^3+5</tt> is represented as F:=[[2x[1],y[1],x[2]^2], [-9y[2],x[1]^2,x[2]^3], [5]]. The zero polynomial <tt>0</tt> is represented as the empty LIST [].</item> |

− | <item>@return: a LIST of | + | <item>@return: a LIST, which is an interreduced set of G.</item> |

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

<example> | <example> | ||

Line 28: | Line 28: | ||

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

<seealso> | <seealso> | ||

− | <see> | + | <see>Use</see> |

<see>NC.SetOrdering</see> | <see>NC.SetOrdering</see> | ||

− | |||

<see>Introduction to CoCoAServer</see> | <see>Introduction to CoCoAServer</see> | ||

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

<types> | <types> | ||

<type>apcocoaserver</type> | <type>apcocoaserver</type> | ||

− | <type> | + | <type>polynomial</type> |

<type>non_commutative</type> | <type>non_commutative</type> | ||

− | |||

</types> | </types> | ||

<key>ncpoly.Interreduction</key> | <key>ncpoly.Interreduction</key> |

## Revision as of 12:29, 26 April 2013

## NC.Interreduction

Interreduction of a LIST of polynomials in a non-commutative polynomial ring.

Note that, given an admissible ordering `Ordering`, a set of non-zero polynomial `G` is called *interreduced* w.r.t. `Ordering` if no element of `Supp(g)` is contained in `LT(G\{g})` for all `g` in `G`.

### Syntax

NC.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 non-commutative polynomial ring (via the command Use) and word ordering (via the function NC.SetOrdering) before calling this function. The default word ordering is the length-lexicographic ordering ("LLEX"). For more information, please check the relevant commands and functions.

@param

*G*: a LIST of non-commutative polynomials. Each polynomial is represented as a LIST of LISTs, and each element in every inner LIST involves only one indeterminate or none (a constant). For example, the polynomial`f=2x[2]y[1]x[2]^2-9y[2]x[1]^2x[2]^3+5`is represented as F:=[[2x[1],y[1],x[2]^2], [-9y[2],x[1]^2,x[2]^3], [5]]. The zero polynomial`0`is represented as the empty LIST [].@return: a LIST, which is an interreduced set of G.

#### Example

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

### See also