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

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(New page: <command> <title>NCo.FindPolynomials</title> <short_description> Find polynomials with specified alphabet (set of indeterminates) from a list of polynomials in monoid rings. </short_descr...)
 
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{{Version|1}}
 
<command>
 
<command>
 
<title>NCo.FindPolynomials</title>
 
<title>NCo.FindPolynomials</title>
 
<short_description>
 
<short_description>
−
Find polynomials with specified alphabet (set of indeterminates) from a list of polynomials in monoid rings.  
+
Find polynomials with specified alphabet (set of indeterminates) from a LIST of non-commutative polynomials.  
 
</short_description>
 
</short_description>
 
<syntax>
 
<syntax>
−
NCo.FindPolynomials(Alphabet:STRING, Polynomials:LIST):LIST
+
NCo.FindPolynomials(Alphabet:STRING, Polys:LIST):LIST
 
</syntax>
 
</syntax>
 
<description>
 
<description>
 
<itemize>
 
<itemize>
 
<item>@param <em>Alphabet</em>: a STRING, which is the specified alphabet.</item>
 
<item>@param <em>Alphabet</em>: a STRING, which is the specified alphabet.</item>
−
<item>@param <em>Polynomials</em>: a LIST of polynomials. Note that 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. Each word in <tt>&lt;X&gt;</tt> is represented as a STRING. For example, the word <tt>xy^2x</tt> is represented as <quotes>xyyx</quotes>, and the identity is represented as the empty string <quotes></quotes>. Thus, the polynomial <tt>F=xy-y+1</tt> in <tt>K&lt;x,y&gt;</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>Polys</em>: a LIST of non-commutative polynomials. Note that 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. Each word in <tt>&lt;X&gt;</tt> is represented as a STRING. For example, the word <tt>xy^2x</tt> is represented as "xyyx", and the identity is represented as the empty string "". Thus, the polynomial <tt>f=xy-y+1</tt> in <tt>K&lt;x,y&gt;</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 polynomials which is the intersection of Polynomials and <tt>K&lt;Alphabet&gt;</tt>.</item>
+
<item>@return: a LIST of polynomials whose indeterminates are in Alphabet.</item>
 
</itemize>
 
</itemize>
 
<example>
 
<example>
−
Polynomials:=[[[1,<quotes>a</quotes>], [1,<quotes>b</quotes>], [1,<quotes>c</quotes>]], [[1,<quotes>b</quotes>]]];
+
Polys:=[[[1,"a"], [1,"b"], [1,"c"]], [[1,"b"]]];
−
NCo.FindPolynomials(<quotes>abc</quotes>, Polynomials);
+
NCo.FindPolynomials("abc", Polys);
  
−
[[[1, <quotes>a</quotes>], [1, <quotes>b</quotes>], [1, <quotes>c</quotes>]], [[1, <quotes>b</quotes>]]]
+
[[[1, "a"], [1, "b"], [1, "c"]], [[1, "b"]]]
 
-------------------------------
 
-------------------------------
−
NCo.FindPolynomials(<quotes>a</quotes>, Polynomials);
+
NCo.FindPolynomials("a", Polys);
  
 
[ ]
 
[ ]
 
-------------------------------
 
-------------------------------
−
NCo.FindPolynomials(<quotes>b</quotes>, Polynomials);
+
NCo.FindPolynomials("b", Polys);
  
−
[[[1, <quotes>b</quotes>]]]
+
[[[1, "b"]]]
 
-------------------------------
 
-------------------------------
−
NCo.FindPolynomials(<quotes>ab</quotes>, Polynomials);
+
NCo.FindPolynomials("ab", Polys);
  
−
[[[1, <quotes>b</quotes>]]]
+
[[[1, "b"]]]
 
-------------------------------
 
-------------------------------
−
NCo.SetX(<quotes>txyz</quotes>);  
+
NCo.SetX("txyz");  
−
NCo.SetOrdering(<quotes>ELIM</quotes>); -- ELIM will eliminate t, x, y, z one after another
+
NCo.SetOrdering("ELIM"); -- ELIM will eliminate t, x, y, z one after another
−
F1 := [[1,<quotes>xx</quotes>], [-1,<quotes>yx</quotes>]];
+
F1 := [[1,"xx"], [-1,"yx"]];
−
F2 := [[1,<quotes>xy</quotes>], [-1,<quotes>ty</quotes>]];
+
F2 := [[1,"xy"], [-1,"ty"]];
−
F3 := [[1,<quotes>xt</quotes>], [-1, <quotes>tx</quotes>]];
+
F3 := [[1,"xt"], [-1, "tx"]];
−
F4 := [[1,<quotes>yt</quotes>], [-1, <quotes>ty</quotes>]];
+
F4 := [[1,"yt"], [-1, "ty"]];
 
G := [F1, F2,F3,F4];  
 
G := [F1, F2,F3,F4];  
 
Gb := NCo.GB(G); -- compute Groebner basis of &lt;G&gt; w.r.t. ELIM
 
Gb := NCo.GB(G); -- compute Groebner basis of &lt;G&gt; w.r.t. ELIM
 
Gb;
 
Gb;
−
NCo.FindPolynomials(<quotes>xyz</quotes>,Gb); -- compute Groebner basis of the intersection of &lt;G&gt; and K&lt;x,y,z&gt; w.r.t. ELIM
+
NCo.FindPolynomials("xyz",Gb); -- compute Groebner basis of the intersection of &lt;G&gt; and K&lt;x,y,z&gt; w.r.t. ELIM
  
−
[[[1, <quotes>xx</quotes>], [2, <quotes>yx</quotes>]], [[1, <quotes>ty</quotes>], [2, <quotes>xy</quotes>]], [[1, <quotes>yt</quotes>], [2, <quotes>xy</quotes>]], [[1, <quotes>tx</quotes>], [2, <quotes>xt</quotes>]],  
+
[[[1, "xx"], [-1, "yx"]], [[1, "ty"], [-1, "xy"]], [[1, "yt"], [-1, "xy"]], [[1, "tx"], [-1, "xt"]],  
−
[[1, <quotes>xyx</quotes>], [2, <quotes>yyx</quotes>]], [[1, <quotes>xyy</quotes>], [2, <quotes>yxy</quotes>]], [[1, <quotes>yxt</quotes>], [2, <quotes>yyx</quotes>]]]
+
[[1, "xyx"], [-1, "yyx"]], [[1, "xyy"], [-1, "yxy"]], [[1, "yxt"], [-1, "yyx"]]]
 
-------------------------------
 
-------------------------------
−
[[[1, <quotes>xx</quotes>], [2, <quotes>yx</quotes>]], [[1, <quotes>xyx</quotes>], [2, <quotes>yyx</quotes>]], [[1, <quotes>xyy</quotes>], [2, <quotes>yxy</quotes>]]]
+
[[[1, "xx"], [-1, "yx"]], [[1, "xyx"], [-1, "yyx"]], [[1, "xyy"], [-1, "yxy"]]]
 
-------------------------------
 
-------------------------------
 
</example>
 
</example>
 
</description>
 
</description>
−
<seealso>
 
−
<see>Introduction to CoCoAServer</see>
 
−
</seealso>
 
 
<types>
 
<types>
−
<type>apcocoaserver</type>
 
 
<type>non_commutative</type>
 
<type>non_commutative</type>
−
<type>polynomial</type>
 
 
</types>
 
</types>
 
<key>gbmr.FindPolynomials</key>
 
<key>gbmr.FindPolynomials</key>
 
<key>NCo.FindPolynomials</key>
 
<key>NCo.FindPolynomials</key>
 
<key>FindPolynomials</key>
 
<key>FindPolynomials</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.FindPolynomials

Find polynomials with specified alphabet (set of indeterminates) from a LIST of non-commutative polynomials.

Syntax

NCo.FindPolynomials(Alphabet:STRING, Polys:LIST):LIST

Description

  • @param Alphabet: a STRING, which is the specified alphabet.

  • @param Polys: a LIST of non-commutative polynomials. Note that 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. Each word in <X> is represented as a STRING. For example, the word xy^2x is represented as "xyyx", and the identity is represented as the empty string "". Thus, the polynomial f=xy-y+1 in K<x,y> is represented as F:=[[1,"xy"], [-1, "y"], [1,""]]. The zero polynomial 0 is represented as the empty LIST [].

  • @return: a LIST of polynomials whose indeterminates are in Alphabet.

Example

Polys:=[[[1,"a"], [1,"b"], [1,"c"]], [[1,"b"]]];
NCo.FindPolynomials("abc", Polys);

[[[1, "a"], [1, "b"], [1, "c"]], [[1, "b"]]]
-------------------------------
NCo.FindPolynomials("a", Polys);

[ ]
-------------------------------
NCo.FindPolynomials("b", Polys);

[[[1, "b"]]]
-------------------------------
NCo.FindPolynomials("ab", Polys);

[[[1, "b"]]]
-------------------------------
NCo.SetX("txyz"); 
NCo.SetOrdering("ELIM"); -- ELIM will eliminate t, x, y, z one after another
F1 := [[1,"xx"], [-1,"yx"]];
F2 := [[1,"xy"], [-1,"ty"]];
F3 := [[1,"xt"], [-1, "tx"]];
F4 := [[1,"yt"], [-1, "ty"]];
G := [F1, F2,F3,F4]; 
Gb := NCo.GB(G); -- compute Groebner basis of &lt;G&gt; w.r.t. ELIM
Gb;
NCo.FindPolynomials("xyz",Gb); -- compute Groebner basis of the intersection of &lt;G&gt; and K&lt;x,y,z&gt; w.r.t. ELIM

[[[1, "xx"], [-1, "yx"]], [[1, "ty"], [-1, "xy"]], [[1, "yt"], [-1, "xy"]], [[1, "tx"], [-1, "xt"]], 
[[1, "xyx"], [-1, "yyx"]], [[1, "xyy"], [-1, "yxy"]], [[1, "yxt"], [-1, "yyx"]]]
-------------------------------
[[[1, "xx"], [-1, "yx"]], [[1, "xyx"], [-1, "yyx"]], [[1, "xyy"], [-1, "yxy"]]]
-------------------------------




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