# ApCoCoA-1:NCo.FindPolynomials

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"]]] -------------------------------