Allow AbstractVectors as input for initial

[YueRen]

Fixes #4025.

Important because functions such as relative_interior_point return PointVectors, and points on a tropical variety or in a Groebner complex are primary candidates for weight vectors.

[benlorenz]

I think most of the changes are unneccessary if you would change the relevant signatures (for groebner_basis and initial) to allow w::AbstractVector{...} instead of w::Vector{...}.

What is still needed is the tropical_semiring(nu)(...), otherwise this leads to some promotion error, but that also happens with Vector{QQFieldElem}.

[YueRen]

I think most of the changes are unneccessary if you would change the relevant signatures (for groebner_basis and initial) to allow w::AbstractVector{...} instead of w::Vector{...}.

What is still needed is the tropical_semiring(nu)(...), otherwise this leads to some promotion error, but that also happens with Vector{QQFieldElem}.

Good point, I've updated the pull request.

[benlorenz]

Somewhat related question that I just noticed while looking at the code here:

Oscar.jl/src/TropicalGeometry/initial.jl

Lines 64 to 70 in 988b607

	if !isnothing(perturbation)
	tropTermsEvaluated = [tropical_semiring(nu)(dot(w,alpha)) for (c,alpha) in zip(coeffs,expvs)]
	tropPolyEvaluated = sum(tropTermsEvaluated)
	termsAttainingValue = findall(isequal(tropPolyEvaluated),tropTermsEvaluated)
	coeffs = coeffs[termsAttainingValue]
	expvs = expvs[termsAttainingValue]
	end

I don't see the contents of perturbation being used at all (except for that isnothing check) ?

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 84.56%. Comparing base (5be006e) to head (8a2755c).
Report is 3 commits behind head on master.

Additional details and impacted files

@@            Coverage Diff             @@
##           master    #4027      +/-   ##
==========================================
+ Coverage   84.06%   84.56%   +0.50%     
==========================================
  Files         597      597              
  Lines       82121    82191      +70     
==========================================
+ Hits        69036    69507     +471     
+ Misses      13085    12684     -401     

Files	Coverage Δ
src/TropicalGeometry/initial.jl	100.00% <100.00%> (+21.73%)	⬆️

... and 41 files with indirect coverage changes

[YueRen]

Somewhat related question that I just noticed while looking at the code here:

Oscar.jl/src/TropicalGeometry/initial.jl

Lines 64 to 70 in 988b607

	if !isnothing(perturbation)
	tropTermsEvaluated = [tropical_semiring(nu)(dot(w,alpha)) for (c,alpha) in zip(coeffs,expvs)]
	tropPolyEvaluated = sum(tropTermsEvaluated)
	termsAttainingValue = findall(isequal(tropPolyEvaluated),tropTermsEvaluated)
	coeffs = coeffs[termsAttainingValue]
	expvs = expvs[termsAttainingValue]
	end

I don't see the contents of perturbation being used at all (except for that isnothing check) ?

Good catch, I think that part is obsolete and can be safely removed. It was originally intended for the tropical traversal algorithm, but
(a) never used
(b) that part can be handled cleaner simply by calling initial twice, imo.

suggestions integrated

[YueRen]

I think the pull-request is ready to be merged, unless @benlorenz has any further comments.