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

(New page: <command> <title>NC.LW</title> <short_description> Leading word of a polynomial in a free monoid ring. </short_description> <syntax> NC.LW(F:LIST):STRING </syntax> <description> <em>Please...) |
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+ | {{Version|1}} | ||

<command> | <command> | ||

<title>NC.LW</title> | <title>NC.LW</title> | ||

<short_description> | <short_description> | ||

− | + | The leading word (or term) of a non-zero polynomial in a non-commutative polynomial ring. | |

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

<syntax> | <syntax> | ||

− | NC.LW(F:LIST): | + | NC.LW(F:LIST):LIST |

+ | NC.LT(F:LIST):LIST | ||

</syntax> | </syntax> | ||

<description> | <description> | ||

<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>ApCoCoA-1:Use|Use</ref>) and word ordering (via the function <ref>ApCoCoA-1:NC.SetOrdering|NC.SetOrdering</ref>) before calling this function. The default word ordering is the length-lexicographic ordering ("LLEX"). For more information, please check the relevant commands and functions. |

<itemize> | <itemize> | ||

− | <item>@param <em>F</em>: a polynomial | + | <item>@param <em>F</em>: a non-zero non-commutative polynomial. 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 | + | <item>@return: a LIST, which is the leading word of F with respect to the current word ordering.</item> |

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

<example> | <example> | ||

− | + | USE QQ[x[1..2]]; | |

− | F:=[[1, | + | F:= [[x[1]^2], [2x[1],x[2]], [3x[2],x[1]],[4x[2]^2]]; -- x[1]^2+2x[1]x[2]+3x[2]x[1]+4x[2]^2 |

− | NC.LW(F); | + | NC.SetOrdering("LLEX"); |

− | + | NC.LW(F); | |

+ | |||

+ | [x[1]^2] | ||

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

− | NC.SetOrdering( | + | -- Done. |

− | NC.LW(F); | + | ------------------------------- |

− | + | NC.SetOrdering("LRLEX"); | |

+ | NC.LW(F); | ||

+ | |||

+ | [x[2]^2] | ||

+ | ------------------------------- | ||

+ | -- Done. | ||

+ | ------------------------------- | ||

+ | NC.SetOrdering("ELIM"); | ||

+ | NC.LW(F); | ||

+ | |||

+ | [x[1]^2] | ||

+ | ------------------------------- | ||

+ | -- Done. | ||

+ | ------------------------------- | ||

+ | NC.SetOrdering("DEGRLEX"); | ||

+ | NC.LT(F); | ||

+ | |||

+ | [x[1]^2] | ||

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

− | |||

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

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

<seealso> | <seealso> | ||

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

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

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

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

<types> | <types> | ||

Line 40: | Line 61: | ||

<key>NC.LW</key> | <key>NC.LW</key> | ||

<key>LW</key> | <key>LW</key> | ||

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

</command> | </command> |

## Latest revision as of 13:35, 29 October 2020

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

## NC.LW

The leading word (or term) of a non-zero polynomial in a non-commutative polynomial ring.

### Syntax

NC.LW(F:LIST):LIST NC.LT(F: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

*F*: a non-zero non-commutative polynomial. 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 the leading word of F with respect to the current word ordering.

#### Example

USE QQ[x[1..2]]; F:= [[x[1]^2], [2x[1],x[2]], [3x[2],x[1]],[4x[2]^2]]; -- x[1]^2+2x[1]x[2]+3x[2]x[1]+4x[2]^2 NC.SetOrdering("LLEX"); NC.LW(F); [x[1]^2] ------------------------------- -- Done. ------------------------------- NC.SetOrdering("LRLEX"); NC.LW(F); [x[2]^2] ------------------------------- -- Done. ------------------------------- NC.SetOrdering("ELIM"); NC.LW(F); [x[1]^2] ------------------------------- -- Done. ------------------------------- NC.SetOrdering("DEGRLEX"); NC.LT(F); [x[1]^2] -------------------------------

### See also