Difference between revisions of "ApCoCoA-1:NCo.FindPolynomials"
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<title>NCo.FindPolynomials</title> | <title>NCo.FindPolynomials</title> | ||
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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> | ||
| − | <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><X></tt> and C is the coefficient of W. Each word in <tt><X></tt> is represented as a STRING. For example, the word <tt>xy^2x</tt> is represented as | + | <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><X></tt> and C is the coefficient of W. Each word in <tt><X></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<x,y></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 whose indeterminates are in Alphabet.</item> | <item>@return: a LIST of polynomials whose indeterminates are in Alphabet.</item> | ||
</itemize> | </itemize> | ||
<example> | <example> | ||
| − | Polys:=[[[1, | + | Polys:=[[[1,"a"], [1,"b"], [1,"c"]], [[1,"b"]]]; |
| − | NCo.FindPolynomials( | + | NCo.FindPolynomials("abc", Polys); |
| − | [[[1, | + | [[[1, "a"], [1, "b"], [1, "c"]], [[1, "b"]]] |
------------------------------- | ------------------------------- | ||
| − | NCo.FindPolynomials( | + | NCo.FindPolynomials("a", Polys); |
[ ] | [ ] | ||
------------------------------- | ------------------------------- | ||
| − | NCo.FindPolynomials( | + | NCo.FindPolynomials("b", Polys); |
| − | [[[1, | + | [[[1, "b"]]] |
------------------------------- | ------------------------------- | ||
| − | NCo.FindPolynomials( | + | NCo.FindPolynomials("ab", Polys); |
| − | [[[1, | + | [[[1, "b"]]] |
------------------------------- | ------------------------------- | ||
| − | NCo.SetX( | + | NCo.SetX("txyz"); |
| − | NCo.SetOrdering( | + | NCo.SetOrdering("ELIM"); -- ELIM will eliminate t, x, y, z one after another |
| − | F1 := [[1, | + | F1 := [[1,"xx"], [-1,"yx"]]; |
| − | F2 := [[1, | + | F2 := [[1,"xy"], [-1,"ty"]]; |
| − | F3 := [[1, | + | F3 := [[1,"xt"], [-1, "tx"]]; |
| − | F4 := [[1, | + | F4 := [[1,"yt"], [-1, "ty"]]; |
G := [F1, F2,F3,F4]; | G := [F1, F2,F3,F4]; | ||
Gb := NCo.GB(G); -- compute Groebner basis of <G> w.r.t. ELIM | Gb := NCo.GB(G); -- compute Groebner basis of <G> w.r.t. ELIM | ||
Gb; | Gb; | ||
| − | NCo.FindPolynomials( | + | NCo.FindPolynomials("xyz",Gb); -- compute Groebner basis of the intersection of <G> and K<x,y,z> w.r.t. ELIM |
| − | [[[1, | + | [[[1, "xx"], [-1, "yx"]], [[1, "ty"], [-1, "xy"]], [[1, "yt"], [-1, "xy"]], [[1, "tx"], [-1, "xt"]], |
| − | [[1, | + | [[1, "xyx"], [-1, "yyx"]], [[1, "xyy"], [-1, "yxy"]], [[1, "yxt"], [-1, "yyx"]]] |
------------------------------- | ------------------------------- | ||
| − | [[[1, | + | [[[1, "xx"], [-1, "yx"]], [[1, "xyx"], [-1, "yyx"]], [[1, "xyy"], [-1, "yxy"]]] |
------------------------------- | ------------------------------- | ||
</example> | </example> | ||
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 <G> w.r.t. ELIM
Gb;
NCo.FindPolynomials("xyz",Gb); -- compute Groebner basis of the intersection of <G> and K<x,y,z> 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"]]]
-------------------------------
