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

Line 5: | Line 5: | ||

</short_description> | </short_description> | ||

<syntax> | <syntax> | ||

− | NCo.FindPolynomials(Alphabet:STRING, | + | 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> | + | <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 <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<x,y></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>@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,<quotes>a</quotes>], [1,<quotes>b</quotes>], [1,<quotes>c</quotes>]], [[1,<quotes>b</quotes>]]]; | |

− | NCo.FindPolynomials(<quotes>abc</quotes>, | + | NCo.FindPolynomials(<quotes>abc</quotes>, Polys); |

[[[1, <quotes>a</quotes>], [1, <quotes>b</quotes>], [1, <quotes>c</quotes>]], [[1, <quotes>b</quotes>]]] | [[[1, <quotes>a</quotes>], [1, <quotes>b</quotes>], [1, <quotes>c</quotes>]], [[1, <quotes>b</quotes>]]] | ||

------------------------------- | ------------------------------- | ||

− | NCo.FindPolynomials(<quotes>a</quotes>, | + | NCo.FindPolynomials(<quotes>a</quotes>, Polys); |

[ ] | [ ] | ||

------------------------------- | ------------------------------- | ||

− | NCo.FindPolynomials(<quotes>b</quotes>, | + | NCo.FindPolynomials(<quotes>b</quotes>, Polys); |

[[[1, <quotes>b</quotes>]]] | [[[1, <quotes>b</quotes>]]] | ||

------------------------------- | ------------------------------- | ||

− | NCo.FindPolynomials(<quotes>ab</quotes>, | + | NCo.FindPolynomials(<quotes>ab</quotes>, Polys); |

[[[1, <quotes>b</quotes>]]] | [[[1, <quotes>b</quotes>]]] |

## Revision as of 14:57, 2 May 2013

## 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,<quotes>a</quotes>], [1,<quotes>b</quotes>], [1,<quotes>c</quotes>]], [[1,<quotes>b</quotes>]]]; NCo.FindPolynomials(<quotes>abc</quotes>, Polys); [[[1, <quotes>a</quotes>], [1, <quotes>b</quotes>], [1, <quotes>c</quotes>]], [[1, <quotes>b</quotes>]]] ------------------------------- NCo.FindPolynomials(<quotes>a</quotes>, Polys); [ ] ------------------------------- NCo.FindPolynomials(<quotes>b</quotes>, Polys); [[[1, <quotes>b</quotes>]]] ------------------------------- NCo.FindPolynomials(<quotes>ab</quotes>, Polys); [[[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 <G> w.r.t. ELIM Gb; NCo.FindPolynomials(<quotes>xyz</quotes>,Gb); -- compute Groebner basis of the intersection of <G> and K<x,y,z> 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>]]] -------------------------------