totalDegreeStartSystem(List) -- construct a start system for the total degree homotopy

Synopsis

Function: totalDegreeStartSystem
Usage:

(S,solsS) = totalDegreeStartSystem T
Inputs:
- T, a list, polynomials of the target system
Outputs:
- a sequence, where S is the list of polynomials in the start system and solsS is the list of start solutions

Description

Given a square target system, constructs a start system for a total degree homotopy together with the total degree (Bezout bound) many start solutions.

For details see: A.J. Sommese, J. Verschelde, and C.W. Wampler, "Introduction to numerical algebraic geometry", in "Solving polynomial equations" (2005), 301--338

i1 : R = CC[x,y,z];

i2 : T = {x^2+y^2-1, x*y^2, x^5+y*z+3};

i3 : totalDegreeStartSystem T

        2       3       5
o3 = ({x  - 1, y  - 1, z  - 1}, {{-1, -.5+.866025*ii, -.809017-.587785*ii},
     ------------------------------------------------------------------------
     {-1, -.5-.866025*ii, -.809017+.587785*ii}, {-1, -.5-.866025*ii, 1}, {-1,
     ------------------------------------------------------------------------
     1, -.809017-.587785*ii}, {1, -.5-.866025*ii, -.809017-.587785*ii}, {-1,
     ------------------------------------------------------------------------
     -.5+.866025*ii, -.809017+.587785*ii}, {-1, -.5-.866025*ii,
     ------------------------------------------------------------------------
     .309017-.951057*ii}, {-1, -.5+.866025*ii, 1}, {1, -.5+.866025*ii,
     ------------------------------------------------------------------------
     -.809017-.587785*ii}, {-1, 1, -.809017+.587785*ii}, {1, -.5-.866025*ii,
     ------------------------------------------------------------------------
     -.809017+.587785*ii}, {-1, -.5+.866025*ii, .309017-.951057*ii}, {-1,
     ------------------------------------------------------------------------
     -.5-.866025*ii, .309017+.951057*ii}, {-1, 1, 1}, {1, -.5-.866025*ii, 1},
     ------------------------------------------------------------------------
     {1, 1, -.809017-.587785*ii}, {1, -.5+.866025*ii, -.809017+.587785*ii},
     ------------------------------------------------------------------------
     {-1, 1, .309017-.951057*ii}, {1, -.5-.866025*ii, .309017-.951057*ii},
     ------------------------------------------------------------------------
     {-1, -.5+.866025*ii, .309017+.951057*ii}, {1, -.5+.866025*ii, 1}, {1, 1,
     ------------------------------------------------------------------------
     -.809017+.587785*ii}, {1, -.5+.866025*ii, .309017-.951057*ii}, {-1, 1,
     ------------------------------------------------------------------------
     .309017+.951057*ii}, {1, -.5-.866025*ii, .309017+.951057*ii}, {1, 1, 1},
     ------------------------------------------------------------------------
     {1, 1, .309017-.951057*ii}, {1, -.5+.866025*ii, .309017+.951057*ii},
     ------------------------------------------------------------------------
     {-1, -.5-.866025*ii, -.809017-.587785*ii}, {1, 1, .309017+.951057*ii}})

o3 : Sequence

Ways to use this method: