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

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+ | {{Version|1}} | ||

<command> | <command> | ||

<title>NC.LW</title> | <title>NC.LW</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 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 ( | + | 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 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>@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> | ||

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USE QQ[x[1..2]]; | 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 | 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( | + | NC.SetOrdering("LLEX"); |

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

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-- Done. | -- Done. | ||

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

− | NC.SetOrdering( | + | NC.SetOrdering("LRLEX"); |

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

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-- Done. | -- Done. | ||

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

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

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

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-- Done. | -- Done. | ||

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

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

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

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

<seealso> | <seealso> | ||

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

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

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

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

<types> | <types> |

## 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