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# decompose(WitnessSet) -- decompose a witness set into irreducibles

## Synopsis

• Function: decompose
• Usage:
Ws = decompose W
• Inputs:
• W, , represents an equidimensional component of a variety
• Outputs:
• Ws, contains irreducible witness sets witness sets, the union of which is W

## Description

Monodromy driven decomposition is followed by the linear trace test.
 i1 : R = CC[x,y] o1 = R o1 : PolynomialRing i2 : F = {x^2+y^2-1, x*y}; i3 : W = first components regeneration F o3 = W o3 : WitnessSet i4 : decompose W o4 = {(dim=0,deg=1), (dim=0,deg=1), (dim=0,deg=1), (dim=0,deg=1)} o4 : List i5 : R = CC[x,y,z] o5 = R o5 : PolynomialRing i6 : sph = (x^2+y^2+z^2-1);  i7 : decompose \ components regeneration {sph*(x-1)*(y-x^2), sph*(y-2)*(z-x^3)} o7 = {{(dim=1,deg=1), (dim=1,deg=1), (dim=1,deg=1), (dim=1,deg=1), ------------------------------------------------------------------------ (dim=1,deg=3)}, {(dim=2,deg=2)}} o7 : List

## Caveat

This function is under development. It can not decompose nonreduced components at the moment. If monodromy breakup algorithm fails to classify some points, the unnclassified points appear as one witness set (that is not marked as irreducible).