# regeneration(List) -- solve a system of polynomial equations with regeneration method

## Synopsis

• Function: regeneration
• Usage:
Ws = regeneration F
• Inputs:
• F, a list, contains polynomials with complex coefficients
• Optional inputs:
• Output => ..., default value Singular
• Software => ..., default value null, specify internal or external software
• Outputs:
• Ws, a list, contains witness sets for equidimensional components of the variety {x|F(x)=0}

## Description

Regeneration is a blackbox method that obtains a numerical description of an algebraic variety. Note that Ws are not necessarily irreducible witness sets; use decompose(WitnessSet) to decompose into irreducibles.
 i1 : R = CC[x,y] o1 = R o1 : PolynomialRing i2 : F = {x^2+y^2-1, x*y}; i3 : regeneration F o3 = a "numerical variety" with components in dim 0: [dim=0,deg=4]-*may be reducible*- o3 : NumericalVariety i4 : R = CC[x,y,z] o4 = R o4 : PolynomialRing i5 : sph = (x^2+y^2+z^2-1);  i6 : regeneration {sph*(x-1)*(y-x^2), sph*(y-2)*(z-x^3)} o6 = a "numerical variety" with components in dim 1: [dim=1,deg=7]-*may be reducible*- dim 2: [dim=2,deg=2]-*may be reducible*- o6 : NumericalVariety

## Caveat

This function is under development. It may not work well if the input represents a nonreduced scheme.The (temporary) option Output can take two values: Regular (default) and Singular. It specifies whether the algorithm attempts to keep singular points.