Difference between revisions of "ApCoCoA-1:NC.NR"
From ApCoCoAWiki
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<title>NC.NR</title> | <title>NC.NR</title> | ||
<short_description> | <short_description> | ||
| − | Normal remainder polynomial with respect to a list of polynomials over a free | + | Normal remainder polynomial with respect to a list of polynomials 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></item> | <item></item> | ||
| − | <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>@param <em>G</em>: a LIST of polynomials in <tt>K<X></tt>.</item> | + | <item>@param <em>G</em>: a LIST of non-zero polynomials in <tt>K<X></tt>.</item> |
| − | <item>@return: a LIST which represents | + | <item>@return: a LIST which represents the normal remainder of <tt>F</tt> w.r.t. <tt>G</tt>.</item> |
</itemize> | </itemize> | ||
<example> | <example> | ||
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NC.NR(F,G); | NC.NR(F,G); | ||
[[1, <quotes>bcb</quotes>], [-1, <quotes>ccc</quotes>], [-1, <quotes>bb</quotes>], [1, <quotes>cb</quotes>], [-1, <quotes></quotes>]] | [[1, <quotes>bcb</quotes>], [-1, <quotes>ccc</quotes>], [-1, <quotes>bb</quotes>], [1, <quotes>cb</quotes>], [-1, <quotes></quotes>]] | ||
| + | |||
------------------------------- | ------------------------------- | ||
NC.SetOrdering(<quotes>ELIM</quotes>); | NC.SetOrdering(<quotes>ELIM</quotes>); | ||
| Line 44: | Line 45: | ||
<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 15:25, 7 June 2012
NC.NR
Normal remainder polynomial with respect to a list of polynomials over a free monoid ring.
Syntax
NC.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 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 [].
@param G: a LIST of non-zero polynomials in K<X>.
@return: a LIST which represents the normal remainder of F w.r.t. G.
Example
NC.SetX(<quotes>abc</quotes>); NC.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]; NC.NR(F,G); [[1, <quotes>bcb</quotes>], [-1, <quotes>ccc</quotes>], [-1, <quotes>bb</quotes>], [1, <quotes>cb</quotes>], [-1, <quotes></quotes>]] ------------------------------- NC.SetOrdering(<quotes>ELIM</quotes>); NC.NR(F,G); [[1, <quotes>bcb</quotes>], [-1, <quotes>bb</quotes>], [1, <quotes>cb</quotes>], [-1, <quotes>ccc</quotes>], [-1, <quotes></quotes>]] -------------------------------
See also
