# ApCoCoA-1:NC.Interreduction

## NC.Interreduction

Interreduction of a LIST of polynomials in a non-commutative polynomial ring.

### Syntax

NC.Interreduction(G:LIST):LIST

### Description

Note that, given a word ordering, a set of non-zero polynomial `G` is called *interreduced* if, for all `g` in `G`, no element of `Supp(g)` is a multiple of any element in `LW{G\{g}}`.

*Please note:* The function(s) explained on this page is/are using the *ApCoCoAServer*. You will have to start the ApCoCoAServer in order to use it/them.

Please set non-commutative polynomial ring (via the command Use) and word ordering (via the function NC.SetOrdering) before calling this function. The default word ordering is the length-lexicographic ordering ("LLEX"). For more information, please check the relevant commands and functions.

@param

*G*: a LIST of non-commutative polynomials. Each polynomial is represented as a LIST of LISTs, and each element in every inner LIST involves only one indeterminate or none (a constant). For example, the polynomial`f=2x[2]y[1]x[2]^2-9y[2]x[1]^2x[2]^3+5`is represented as F:=[[2x[1],y[1],x[2]^2], [-9y[2],x[1]^2,x[2]^3], [5]]. The zero polynomial`0`is represented as the empty LIST [].@return: a LIST, which is an interreduced set of G.

#### Example

USE QQ[x[1..2],y[1..2]]; NC.SetOrdering(<quotes>LLEX</quotes>); F1:= [[x[1],y[1],x[2]^2], [-9y[2],x[1]^2,x[2]^3],[5]]; -- x[1]y[1]x[2]^2-9y[2]x[1]^2x[2]^3+5 F2:= [[y[1],x[2]^2], [y[2],x[2]^2]]; -- y[1]x[2]^2+y[2]x[2]^2 F3:= [[x[1],y[1]],[x[2]]]; -- x[1]y[1]+x[2] NC.Interreduction([F1,F2,F3]); [[[y[2], x[1]^2, x[2]^3], [1/9x[1], y[2], x[2]^2], [-5/9]], [[y[1], x[2]^2], [y[2], x[2]^2]], [[x[1], y[1]], [x[2]]]] -------------------------------

### See also