# ApCoCoA-1:NCo.FindPolynomials

## NCo.FindPolynomials

Find polynomials with specified alphabet (set of indeterminates) from a list of polynomials in monoid rings.

### Syntax

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

### Description

@param

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

*Polynomials*: 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`<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 which is the intersection of Polynomials and

`K<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 <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>]]] -------------------------------

### See also