# cascade -- runs a cascade of homotopies to get witness sets for the variety

## Synopsis

• Usage:
cascade L
• Inputs:
• L, a list, of polynomials
• Optional inputs:
• StartDimension => ..., default value -1, Option to specify the dimension to begin searching for positive dimensional components
• Verbose => ..., default value false, option to specify whether additional output is wanted
• Outputs:
• , containing the witness sets for each pure-dimensional variety contained in zero set of L.
• Consequences:
• Writes the system to temporary file
• Invokes the command phc -c (with option 0)
• Stores output of phc in temporary file
• Parses and outputs the solutions.

## Description

Given a list of generators of an ideal I, this function creates an embedding and then runs a cascade of homotopies. The output is a NumericalVariety that contains a WitnessSet for each pure dimensional variety contained in V(I).

 i1 : R = CC[x,y,z]; i2 : L = { z*(x+y), z*(x-y) }; i3 : WitSets = cascade(L) o3 = WitSets o3 : NumericalVariety i4 : W=first WitSets#2 o4 = W o4 : WitnessSet

The function cascade extends the ring of the inputted system with slack variables beginning with zz. Each witness set in contains the equations, points, and slices of the embedded system.

 i5 : W#Equations o5 = ideal ((- .380139 + .924929*ii)zz1 + (.547606 + .836736*ii)zz2 - x*z - ------------------------------------------------------------------------ y*z, (.45571 - .890128*ii)zz1 + (- .663772 - .747935*ii)zz2 - x*z + y*z, ------------------------------------------------------------------------ - zz1 + (.537862 + .843033*ii)zz2) o5 : Ideal of CC {x, y, z, zz1, zz2} 53 i6 : W#Points o6 = {{.148619-.161802*ii, .45886-1.03272*ii, 3.21921e-32+7.65273e-33*ii, -2.28184e-32-4.87389e-33*ii, -1.6382e-32+1.66152e-32*ii}} o6 : VerticalList i7 : W#Slice o7 = | .205377+.978683ii .673112+.739541ii .943327-.331864ii | -.759495+.650513ii -.89022-.45553ii -.972487+.232959ii ------------------------------------------------------------------------ -.911067-.412259ii .497812+.867285ii -.883724+.468009ii | -.595572-.803302ii -.236638-.971598ii .871298-.490754ii | 2 6 o7 : Matrix CC <--- CC 53 53

## Caveat

Coefficient ring of the polynomial system must be of type ComplexField.

## Ways to use cascade :

• "cascade(List)"

## For the programmer

The object cascade is .