Automorphisms

Automorphisms of a parametric polynomial system are birational maps that fix the parameters of a polynomial system.

Interpolating automorphisms

DecomposingPolynomialSystems.automorphisms — Function
automorphisms(F::ParametricSystem; degree_bound=1, param_dep=true, kwargs...)

Given a polynomial system F returns the group of automorphisms of F (symmetries that fix the parameters). The keyword argument degree_bound is used to set the upper bound for the degrees of numerator and denominator polynomials in expressions for the symmetries. The param_dep keyword argument specifies whether to consider functions of the symmetries to be dependent on the parameters of F.

julia> @var x[1:2] p[1:2];

julia> F = ParametricSystem([x[1]^2 - x[2]^2 - p[1], 2*x[1]*x[2] - p[2]]; unknowns=x, parameters=p);

julia> automorphisms(F; degree_bound=1, param_dep=false)
AutomorphismGroup of order 4
 structure: C2 x C2
 action:
  1st map:
   x₁ ↦ x₁
   x₂ ↦ x₂
  2nd map:
   x₁ ↦ -x₁
   x₂ ↦ -x₂
  3rd map:
   x₁ ↦ im*x₂
   x₂ ↦ -im*x₁
  4th map:
   x₁ ↦ -im*x₂
   x₂ ↦ im*x₁
source

AutomorphismGroup