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# 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:
• , 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

• track -- track a linear segment homotopy given start and target system
• solveSystem -- solve a system of polynomial equations