Difference between revisions of "ApCoCoA-1:NCo.Interreduction"
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<command> | <command> | ||
<title>NCo.Interreduction</title> | <title>NCo.Interreduction</title> | ||
<short_description> | <short_description> | ||
Interreduce a LIST of polynomials in a free monoid ring. | Interreduce a LIST of polynomials in a free monoid ring. | ||
| − | |||
| − | |||
</short_description> | </short_description> | ||
<syntax> | <syntax> | ||
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</syntax> | </syntax> | ||
<description> | <description> | ||
| + | Note that, given a word ordering <tt>Ordering</tt>, a set of non-zero polynomials <tt>G</tt> is called <em>interreduced</em> with respect to <tt>Ordering</tt> if no element of <tt>Supp(g)</tt> is contained in the leading word ideal <tt>LW(G\{g})</tt> for all <tt>g</tt> in <tt>G</tt>. | ||
| + | <par/> | ||
<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 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>@param <em>G</em>: a LIST of polynomials 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>G</em>: a LIST of polynomials 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>@return: a LIST of interreduced polynomials with respect to the current word ordering.</item> | <item>@return: a LIST of interreduced polynomials with respect to the current word ordering.</item> | ||
</itemize> | </itemize> | ||
<example> | <example> | ||
| − | NCo.SetX( | + | NCo.SetX("abc"); |
| − | NCo.SetOrdering( | + | NCo.SetOrdering("ELIM"); |
| − | G:=[[[1, | + | G:=[[[1,"ba"]], [[1,"b"],[1,""]], [[1,"c"]]]; |
NCo.Interreduction(G); | NCo.Interreduction(G); | ||
| − | [[[1, | + | [[[1, "a"]], [[1, "b"], [1, ""]], [[1, "c"]]] |
------------------------------- | ------------------------------- | ||
</example> | </example> | ||
</description> | </description> | ||
<seealso> | <seealso> | ||
| − | <see>NCo.LW</see> | + | <see>ApCoCoA-1:NCo.LW|NCo.LW</see> |
| − | <see>NCo.SetFp</see> | + | <see>ApCoCoA-1:NCo.SetFp|NCo.SetFp</see> |
| − | <see>NCo.SetOrdering</see> | + | <see>ApCoCoA-1:NCo.SetOrdering|NCo.SetOrdering</see> |
| − | <see>NCo.SetX</see> | + | <see>ApCoCoA-1:NCo.SetX|NCo.SetX</see> |
| − | <see>Introduction to CoCoAServer</see> | + | <see>ApCoCoA-1:Introduction to CoCoAServer|Introduction to CoCoAServer</see> |
</seealso> | </seealso> | ||
<types> | <types> | ||
| Line 43: | Line 44: | ||
<key>NCo.Interreduction</key> | <key>NCo.Interreduction</key> | ||
<key>Interreduction</key> | <key>Interreduction</key> | ||
| − | <wiki-category>Package_gbmr</wiki-category> | + | <wiki-category>ApCoCoA-1:Package_gbmr</wiki-category> |
</command> | </command> | ||
Latest revision as of 13:39, 29 October 2020
| This article is about a function from ApCoCoA-1. |
NCo.Interreduction
Interreduce a LIST of polynomials in a free monoid ring.
Syntax
NCo.Interreduction(G:LIST):LIST
Description
Note that, given a word ordering Ordering, a set of non-zero polynomials G is called interreduced with respect to Ordering if no element of Supp(g) is contained in the leading word ideal LW(G\{g}) for all g in G.
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 G: a LIST of polynomials 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 [].
@return: a LIST of interreduced polynomials with respect to the current word ordering.
Example
NCo.SetX("abc");
NCo.SetOrdering("ELIM");
G:=[[[1,"ba"]], [[1,"b"],[1,""]], [[1,"c"]]];
NCo.Interreduction(G);
[[[1, "a"]], [[1, "b"], [1, ""]], [[1, "c"]]]
-------------------------------
See also
