solveSystem(List) -- solve a system of polynomial equations

Synopsis

• Function: solveSystem

• Usage:

  s = solveSystem F

• Inputs:
  • F, a list, contains polynomials with complex coefficients
• Optional inputs:
  • CorrectorTolerance => ..., default value null, options for core functions of Numerical Algebraic Geometry
  • EndZoneFactor => ..., default value null, options for core functions of Numerical Algebraic Geometry
  • gamma => ..., default value null, options for core functions of Numerical Algebraic Geometry
  • InfinityThreshold => ..., default value null, options for core functions of Numerical Algebraic Geometry
  • maxCorrSteps => ..., default value null, options for core functions of Numerical Algebraic Geometry
  • Normalize => ..., default value null, options for core functions of Numerical Algebraic Geometry
  • numberSuccessesBeforeIncrease => ..., default value null, options for core functions of Numerical Algebraic Geometry
  • PostProcess => ..., default value true, specifies whether to postprocess the solutions
  • Precision => ..., default value null, options for core functions of Numerical Algebraic Geometry
  • Predictor => ..., default value null, options for core functions of Numerical Algebraic Geometry
  • Projectivize => ..., default value null, options for core functions of Numerical Algebraic Geometry
  • ResidualTolerance => ..., default value null, options for core functions of Numerical Algebraic Geometry
  • SingularConditionNumber => ..., default value null, options for core functions of Numerical Algebraic Geometry
  • Software => ..., default value null, specify internal or external software
  • stepIncreaseFactor => ..., default value null, options for core functions of Numerical Algebraic Geometry
  • tDegree => ..., default value null, options for core functions of Numerical Algebraic Geometry
  • tStep => ..., default value null, options for core functions of Numerical Algebraic Geometry
  • tStepMin => ..., default value null, options for core functions of Numerical Algebraic Geometry
• Outputs:
  • s, a list, contains all complex solutions to the system F=0

Description

i1 : R = CC[x,y];

i2 : F = {x^2+y^2-1, x*y};

i3 : solveSystem F 

o3 = {{3.02251e-20+2.05294e-20*ii, -1}, {-3.02251e-20-2.05294e-20*ii, 1}, {1,
     ------------------------------------------------------------------------
     8.30314e-21+4.44918e-20*ii}, {-1, -8.30314e-21-4.44918e-20*ii}}

o3 : List

i4 : R = CC[x,y];

i5 : F = {x^2+y^2-1, x*y, x*(y+1)};

i6 : solveSystem F 

o6 = {{5.40508e-17+2.78984e-17*ii, -1}, {3.45943e-17+4.909e-17*ii, 1}}

o6 : List

The system is assumed to have finitely many solutions. If it is not square (number of equations = number of variables), squareUp is applied and solutions to the original system are then picked out from the resulting (larger) set of solutions.

The output (produced by track with default options) contains all points obtained at the end of homotopy paths when tracking starting at the totalDegreeStartSystem. In particular, this means that solving a system that has fewer than Bezout bound many solutions will produce points that are not marked as regular. See track for detailed examples.