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

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<title>NC.LC</title> | <title>NC.LC</title> | ||

<short_description> | <short_description> | ||

− | Leading coefficient of a polynomial over a free | + | Leading coefficient of a polynomial over a free monoid ring. |

</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 coefficient field <tt>K</tt>, alphabet (or indeterminates) <tt>X</tt> and 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>NC.SetFp</ref>, <ref>NC.SetX</ref> and <ref>NC.SetOrdering</ref>, respectively, before calling the function. Default coefficient field is <tt>Q</tt>. Default ordering is length-lexicographic ordering (<quotes>LLEX</quotes>). For more information, please check the relevant functions. |

<itemize> | <itemize> | ||

− | <item>@param <em>F</em>: a polynomial in <tt>K<X></tt>. Each polynomial is represented as a LIST of | + | <item>@param <em>F</em>: a polynomial in <tt>K<X></tt>. Each polynomial is represented as a LIST of monomials, which are pairs 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 an empty LIST [].</item> |

− | <item>@return: a INT or RAT which represents | + | <item>@return: a INT or RAT which represents the leading coefficient of F. If <tt>F=0</tt>, the function returns <tt>0</tt>.</item> |

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

<example> | <example> | ||

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NC.LC(F); | NC.LC(F); | ||

2 | 2 | ||

+ | |||

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

NC.SetOrdering(<quotes>LLEX</quotes>); | NC.SetOrdering(<quotes>LLEX</quotes>); | ||

NC.LC(F); | NC.LC(F); | ||

4 | 4 | ||

+ | |||

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

NC.LC([]); -- 0 polynomial | NC.LC([]); -- 0 polynomial | ||

0 | 0 | ||

+ | |||

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

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

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<see>NC.GB</see> | <see>NC.GB</see> | ||

<see>NC.HF</see> | <see>NC.HF</see> | ||

+ | <see>NC.Interreduction</see> | ||

<see>NC.Intersection</see> | <see>NC.Intersection</see> | ||

<see>NC.IsGB</see> | <see>NC.IsGB</see> |

## Revision as of 15:00, 7 June 2012

## NC.LC

Leading coefficient of a polynomial over a free monoid ring.

### Syntax

NC.LC(F:LIST):INT or RAT

### 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 NC.SetFp, NC.SetX and NC.SetOrdering, respectively, before calling the function. Default coefficient field is `Q`. Default ordering is length-lexicographic ordering ("LLEX"). For more information, please check the relevant functions.

@param

*F*: a polynomial in`K<X>`. Each polynomial is represented as a LIST of monomials, which are pairs 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 an empty LIST [].@return: a INT or RAT which represents the leading coefficient of F. If

`F=0`, the function returns`0`.

#### Example

NC.SetX(<quotes>abc</quotes>); F:=[[1,<quotes>ab</quotes>],[2,<quotes>aa</quotes>],[3,<quotes>bb</quotes>],[4,<quotes>bab</quotes>]]; NC.SetOrdering(<quotes>ELIM</quotes>); NC.LC(F); 2 ------------------------------- NC.SetOrdering(<quotes>LLEX</quotes>); NC.LC(F); 4 ------------------------------- NC.LC([]); -- 0 polynomial 0 -------------------------------

### See also