tropical determinant via Hungarian method

src/TropicalGeometry/matrix.jl

@@ -14,7 +14,9 @@
 @doc raw"""
     det(A::Generic.MatSpaceElem{<: TropicalSemiringElem})
 
-Return the tropical determinant of `A`.
+Return the tropical determinant of `A`.  That is, this function evaluates the tropicalization of the ordinary determinant considered as a multivariate polynomial at `A`.
+
+That computation is equivalent to solving a linear assignment problem from combinatorial optimization.  The implementation employs the Hungarian method, which is polynomial time.  See Chapter 3 in [Jos21](@cite).
 
 !!! note
     This function effectively overwrites the `det` command for tropical matrices.  This means that functions like `minors` will use the tropical determinant when used on a tropical matrix.
@@ -30,12 +32,9 @@ julia> det(A)
 ```
 """
 function det(A::Generic.MatSpaceElem{R}) where {R<:Union{TropicalSemiringElem,MPolyRingElem{<:TropicalSemiringElem},PolyRingElem{<:TropicalSemiringElem}}}
-    detA = zero(base_ring(A))
-    nrows(A)!=ncols(A) && return detA # return tropical zero if matrix not square
-    for sigma in AbstractAlgebra.SymmetricGroup(nrows(A)) # otherwise follow Leibniz Formula
-        detA += prod([ A[i,sigma[i]] for i in 1:nrows(A) ])
-    end
-    return detA
+    T = base_ring(A)
+    nrows(A)!=ncols(A) && return zero(T) # return tropical zero if matrix not square
+    return T(Polymake.tropical.tdet(A))
 end
 
 function det(A::Matrix{R}) where {R<:Union{TropicalSemiringElem,MPolyRingElem{<:TropicalSemiringElem},PolyRingElem{<:TropicalSemiringElem}}}