# Witness Sets

A WitnessSet $W$ is the basic data structure to work with positive dimensional solution sets $V(F)$ of a polynomial system $F$. The general idea is to intersect $V(F)$ with an (affine) linear space $L$ such that the intersection $V(F) ∩ L$ consists of only finitely many points (witnesses). Over the complex numbers the number of points is independent of the linear space $L$ and called the degree of $V(F)$.

To compute a WitnessSet call witness_set.

HomotopyContinuation.witness_setFunction
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
source

To obtain information about a WitnessSet the following functions are provided.

To test for completeness of a WitnessSet you can perform a trace_test

HomotopyContinuation.trace_testFunction
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

source
• LRS18Leykin, Anton, Jose Israel Rodriguez, and Frank Sottile. "Trace test." Arnold Mathematical Journal 4.1 (2018): 113-125.