Difference between revisions of "ApCoCoA-1:NCo.NR"
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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>ApCoCoA-1:NCo.SetFp|NCo.SetFp</ref>, <ref>ApCoCoA-1:NCo.SetX|NCo.SetX</ref> and <ref>ApCoCoA-1:NCo.SetOrdering|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 ( | + | 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>ApCoCoA-1:NCo.SetFp|NCo.SetFp</ref>, <ref>ApCoCoA-1:NCo.SetX|NCo.SetX</ref> and <ref>ApCoCoA-1:NCo.SetOrdering|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 ("LLEX"). 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 monomials, which are LISTs 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, | + | <item>@param <em>F</em>: a polynomial in <tt>K<X></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><X></tt> and C is the coefficient of W. For example, the polynomial <tt>f=xy-y+1</tt> is represented as F:=[[1,"xy"], [-1, "y"], [1,""]]. 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<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 the normal remainder of <tt>F</tt> with respect to <tt>G</tt>.</item> | <item>@return: a LIST which represents the normal remainder of <tt>F</tt> with respect to <tt>G</tt>.</item> | ||
</itemize> | </itemize> | ||
<example> | <example> | ||
| − | NCo.SetX( | + | NCo.SetX("abc"); |
NCo.RingEnv(); | NCo.RingEnv(); | ||
Coefficient ring : Q | Coefficient ring : Q | ||
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Ordering : LLEX | Ordering : LLEX | ||
------------------------------- | ------------------------------- | ||
| − | F:=[[1, | + | F:=[[1,"ab"],[1,"aca"],[1,"bb"],[1,"bab"],[1,""]]; |
| − | F1 := [[1, | + | F1 := [[1,"a"],[1,"c"]]; |
| − | F2 := [[1, | + | F2 := [[1,"b"],[1,"ba"]]; |
G:=[F1,F2]; | G:=[F1,F2]; | ||
NCo.NR(F,G); | NCo.NR(F,G); | ||
| − | [[-1, | + | [[-1,"bcb"], [1,"ccc"], [1,"bb"], [-1,"cb"], [1,""]] |
------------------------------- | ------------------------------- | ||
| − | NCo.SetOrdering( | + | NCo.SetOrdering("ELIM"); |
NCo.NR(F,G); | NCo.NR(F,G); | ||
| − | [[-1, | + | [[-1,"bcb"], [1,"bb"], [-1,"cb"], [1,"ccc"], [1,""]] |
------------------------------- | ------------------------------- | ||
</example> | </example> | ||
Latest revision as of 13:43, 29 October 2020
| This article is about a function from ApCoCoA-1. |
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("abc");
NCo.RingEnv();
Coefficient ring : Q
Alphabet : abc
Ordering : LLEX
-------------------------------
F:=[[1,"ab"],[1,"aca"],[1,"bb"],[1,"bab"],[1,""]];
F1 := [[1,"a"],[1,"c"]];
F2 := [[1,"b"],[1,"ba"]];
G:=[F1,F2];
NCo.NR(F,G);
[[-1,"bcb"], [1,"ccc"], [1,"bb"], [-1,"cb"], [1,""]]
-------------------------------
NCo.SetOrdering("ELIM");
NCo.NR(F,G);
[[-1,"bcb"], [1,"bb"], [-1,"cb"], [1,"ccc"], [1,""]]
-------------------------------
See also
