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goodInitialPair -- a good (conjectured by Shub and Smale) initial pair

Synopsis

Usage:
(S,sol) = goodInitialPair T
Inputs:
- T, a list of polynomials
Outputs:
- S, list of polynomials
- sol, a list containing (one) solution of S
Optional inputs:
- GeneralPosition => default value false, make a random unitary change of coordinates

Description

Generates a start system S that is conjectured to have good complexity when used in linear homotopy with target system T leading to one solution.

i1 : T = randomSd {2,3};

i2 : (S,solsS) = goodInitialPair T

                                 2
o2 = ({1.41421x1*x3, 1.73205x2*x3 }, {{0, 0, 1}})

o2 : Sequence

i3 : M = track(S,T,solsS,gamma=>0.6+0.8*ii,Software=>M2)

o3 = {{{.26568-.427252*ii, .453086-.286339*ii, .649656+.193752*ii}, #steps =>
     ------------------------------------------------------------------------
     11, status  => REGULAR, last t => 1, cond#^{-1} => .47794}}

o3 : List

Ways to use `goodInitialPair` :

goodInitialPair(List)