Witness Sets

WitnessSet(F, L, S)

Store solutions S of the polynomial system F(x) = L(x) = 0 into a witness set.

witness_set(F; codim = nvariables(F) - length(F), dim = nothing, options...)

Compute a WitnessSet for F in the given dimension (resp. codimension) by sampling a random (affine) linear subspace. After constructing the system this calls solve with the provided options.

witness_set(F, L; options...)

Compute WitnessSet for F and the (affine) linear subspace L.

witness_set(W::WitnessSet, L; options...)

Compute a new WitnessSet with the (affine) linear subspace L by moving the linear subspace stored in W to L.

Example

julia> @var x y;
julia> F = System([x^2 + y^2 - 5], [x, y])
System of length 1
 2 variables: x, y

 -5 + x^2 + y^2

julia> W = witness_set(F)
Witness set for dimension 1 of degree 2

solutions(W::WitnessSet)

Get the solutions stored in W.

results(W::WitnessSet)

Get the results stored in W.

system(W::WitnessSet)

Get the system stored in W.

linear_subspace(W::WitnessSet)

Get the linear subspace stored in W.

dim(W::WitnessSet)

The dimension of the algebraic set encoded by the witness set.

codim(W::WitnessSet)

The dimension of the algebraic set encoded by the witness set.

degree(W::WitnessSet)

Returns the degree of the witness set W. This equals the number of solutions stored.

trace_test(W::WitnessSet; options...)

Performs a trace test ^[LRS18] to verify whether the given witness set W is complete. Returns the trace of the witness set which should be theoretically be 0 if W is complete. Due to floating point arithmetic this is not the case, thus is has to be manually checked that the trace is sufficiently small. Returns nothing if the trace test failed due to path tracking failures. The options are the same as for calls to witness_set.

julia> @var x y;
julia> F = System([x^2 + y^2 - 5], [x, y])
System of length 1
 2 variables: x, y

 -5 + x^2 + y^2

julia> W = witness_set(F)
Witness set for dimension 1 of degree 2

julia> trace = trace_test(W)
9.981960497718987e-16

APA