Difference between revisions of "ApCoCoA-1:NCo.NR"

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(New page: <command> <title>NCo.NR</title> <short_description> Normal remainder of polynomial with respect to a list of polynomials in a free monoid ring. </short_description> <syntax> NCo.NR(F:LIST,...)
 
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<title>NCo.NR</title>
 
<title>NCo.NR</title>
 
<short_description>
 
<short_description>
Normal remainder of polynomial with respect to a list of polynomials in a free monoid ring.
+
The normal remainder of a polynomial with respect to a LIST of polynomials in 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 <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 calling the function. The default coefficient field is <tt>Q</tt>. The default ordering is length-lexicographic ordering (<quotes>LLEX</quotes>). For more information, please check the relevant functions.
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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 (<quotes>LLEX</quotes>). For more information, please check the relevant functions.
 
<itemize>
 
<itemize>
 
<item></item>
 
<item></item>
<item>@param <em>F</em>: a polynomial in <tt>K&lt;X&gt;</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>&lt;X&gt;</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 the empty LIST [].</item>
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<item>@param <em>F</em>: a polynomial in <tt>K&lt;X&gt;</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>&lt;X&gt;</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 the empty LIST [].</item>
 
<item>@param <em>G</em>: a LIST of non-zero polynomials in <tt>K&lt;X&gt;</tt>.</item>
 
<item>@param <em>G</em>: a LIST of non-zero polynomials in <tt>K&lt;X&gt;</tt>.</item>
<item>@return: a LIST which represents the normal remainder of <tt>F</tt> w.r.t. <tt>G</tt>.</item>
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<item>@return: a LIST which represents the normal remainder of <tt>F</tt> with respect to <tt>G</tt>.</item>
 
</itemize>
 
</itemize>
 
<example>
 
<example>

Revision as of 16:46, 29 April 2013

NCo.NR

The normal remainder of a polynomial with respect to a LIST of polynomials in a free monoid ring.

Syntax

NCo.NR(F:LIST, 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 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 F: a polynomial 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 [].

  • @param G: a LIST of non-zero polynomials in K<X>.

  • @return: a LIST which represents the normal remainder of F with respect to G.

Example

NCo.SetX(<quotes>abc</quotes>);
NCo.RingEnv();
Coefficient ring : Q
Alphabet : abc
Ordering : LLEX
-------------------------------
F:=[[1,<quotes>ab</quotes>],[1,<quotes>aca</quotes>],[1,<quotes>bb</quotes>],[1,<quotes>bab</quotes>],[1,<quotes></quotes>]];
F1 := [[1,<quotes>a</quotes>],[1,<quotes>c</quotes>]]; 
F2 := [[1,<quotes>b</quotes>],[1,<quotes>ba</quotes>]];
G:=[F1,F2];
NCo.NR(F,G);
[[1, <quotes>bcb</quotes>], [-1, <quotes>ccc</quotes>], [-1, <quotes>bb</quotes>], [1, <quotes>cb</quotes>], [-1, <quotes></quotes>]]
-------------------------------
NCo.SetOrdering(<quotes>ELIM</quotes>);
NCo.NR(F,G);
[[1, <quotes>bcb</quotes>], [-1, <quotes>bb</quotes>], [1, <quotes>cb</quotes>], [-1, <quotes>ccc</quotes>], [-1, <quotes></quotes>]]
-------------------------------

See also

NCo.SetFp

NCo.SetOrdering

NCo.SetX

Introduction to CoCoAServer