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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<title>NCo.FindPolynomials</title>
 
<title>NCo.FindPolynomials</title>
 
<short_description>
 
<short_description>
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Find polynomials with specified alphabet (set of indeterminates) from a list of polynomials in monoid rings.  
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Find polynomials with specified alphabet (set of indeterminates) from a LIST of non-commutative polynomials.  
 
</short_description>
 
</short_description>
 
<syntax>
 
<syntax>
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<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>
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<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>
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<item>@param <em>Polynomials</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 <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>
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<item>@return: a LIST of polynomials which is the intersection of Polynomials and <tt>K&lt;Alphabet&gt;</tt>.</item>
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<item>@return: a LIST of polynomials whose indeterminates are in Alphabet.</item>
 
</itemize>
 
</itemize>
 
<example>
 
<example>
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</example>
 
</example>
 
</description>
 
</description>
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<seealso>
 
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<see>Introduction to CoCoAServer</see>
 
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</seealso>
 
 
<types>
 
<types>
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<type>apcocoaserver</type>
 
 
<type>non_commutative</type>
 
<type>non_commutative</type>
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<type>polynomial</type>
 
 
</types>
 
</types>
 
<key>gbmr.FindPolynomials</key>
 
<key>gbmr.FindPolynomials</key>

Revision as of 16:27, 30 April 2013

NCo.FindPolynomials

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

Syntax

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

Description

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

  • @param Polynomials: 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

Polynomials:=[[[1,<quotes>a</quotes>], [1,<quotes>b</quotes>], [1,<quotes>c</quotes>]], [[1,<quotes>b</quotes>]]];
NCo.FindPolynomials(<quotes>abc</quotes>, Polynomials);

[[[1, <quotes>a</quotes>], [1, <quotes>b</quotes>], [1, <quotes>c</quotes>]], [[1, <quotes>b</quotes>]]]
-------------------------------
NCo.FindPolynomials(<quotes>a</quotes>, Polynomials);

[ ]
-------------------------------
NCo.FindPolynomials(<quotes>b</quotes>, Polynomials);

[[[1, <quotes>b</quotes>]]]
-------------------------------
NCo.FindPolynomials(<quotes>ab</quotes>, Polynomials);

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

[[[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, <quotes>xyx</quotes>], [2, <quotes>yyx</quotes>]], [[1, <quotes>xyy</quotes>], [2, <quotes>yxy</quotes>]], [[1, <quotes>yxt</quotes>], [2, <quotes>yyx</quotes>]]]
-------------------------------
[[[1, <quotes>xx</quotes>], [2, <quotes>yx</quotes>]], [[1, <quotes>xyx</quotes>], [2, <quotes>yyx</quotes>]], [[1, <quotes>xyy</quotes>], [2, <quotes>yxy</quotes>]]]
-------------------------------




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