# bertiniTrackHomotopy -- a main method to track using a user-defined homotopy

## Synopsis

• Usage:
S0=bertiniTrackHomotopy(t, H, S1)
• Inputs:
• t, , a path variable
• H, a list, a list polynomials that define the homotopy with respect to the path variable
• S1, a list, a list of solutions to the start system
• Optional inputs:
• BertiniInputConfiguration (missing documentation) => ..., default value {},
• IsProjective => ..., default value -1, optional argument to specify whether to use homogeneous coordinates
• Verbose => ..., default value false, Option to silence additional output
• Outputs:
• S0, a list, a list of solutions to the target system

## Description

This method calls Bertini to track a user-defined homotopy. The user needs to specify the homotopy H, the path variable t, and a list of start solutions S1. Bertini (1) writes the homotopy and start solutions to temporary files, (2) invokes Bertini's solver with configuration keyword UserHomotopy => 1 in the affine case and UserHomotopy => 2 in the projective situation, (3) stores the output of Bertini in a temporary file, and (4) parses a machine readable file to output a list of solutions.

 i1 : R = CC[x,t]; -- include the path variable in the ring i2 : H = { (x^2-1)*t + (x^2-2)*(1-t)}; i3 : sol1 = point {{1}}; i4 : sol2 = point {{-1}}; i5 : S1= { sol1, sol2 };--solutions to H when t=1 i6 : S0 = bertiniTrackHomotopy (t, H, S1) --solutions to H when t=0 o6 = {{-1.41421}, {1.41421}} o6 : List i7 : peek S0_0 o7 = Point{ConditionNumber => 1 } Coordinates => {-1.41421} CycleNumber => 1 FunctionResidual => 4.44089e-16 LastT => .0015625 MaximumPrecision => 52 Multiplicity => 1 NewtonResidual => 0 SolutionNumber => -1 SolutionStatus => Regular

In the previous example, we solved $x^2-2$ by moving from $x^2-1$ with a linear homotopy. Bertini tracks homotopies starting at $t=1$ and ending at $t=0$. Final solutions are of the type Point.

 i8 : R=CC[x,y,t]; -- include the path variable in the ring i9 : f1=(x^2-y^2); i10 : f2=(2*x^2-3*x*y+5*y^2); i11 : H = { f1*t + f2*(1-t)}; --H is a list of polynomials in x,y,t i12 : sol1= point{{1,1}}--{{x,y}} coordinates o12 = sol1 o12 : Point i13 : sol2= point{{ -1,1}} o13 = sol2 o13 : Point i14 : S1={sol1,sol2}--solutions to H when t=1 o14 = {sol1, sol2} o14 : List i15 : S0=bertiniTrackHomotopy(t, H, S1, IsProjective=>1) --solutions to H when t=0 o15 = {{-.947782-.710437*ii, .11122-.740833*ii}, {-.847231+1.44261*ii, ----------------------------------------------------------------------- .549043+.904502*ii}} o15 : List

Variables must begin with a letter (lowercase or capital) and can only contain letters, numbers, underscores, and square brackets.

## Ways to use bertiniTrackHomotopy :

• "bertiniTrackHomotopy(RingElement,List,List)"

## For the programmer

The object bertiniTrackHomotopy is .