tropical determinant via Hungarian method (#3943)

src/TropicalGeometry/matrix.jl

@@ -12,9 +12,11 @@
 ################################################################################
 
 @doc raw"""
-    det(A::Generic.MatSpaceElem{<: TropicalSemiringElem})
+    det(A::MatrixElem{<: 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.
@@ -29,15 +31,13 @@ julia> det(A)
 (5)
 ```
 """
-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
+function det(A::MatrixElem{<:TropicalSemiringElem})
+  @req nrows(A) == ncols(A) "Non-square matrix"
+  T = base_ring(A)
+  return T(Polymake.tropical.tdet(A))
 end
 
-function det(A::Matrix{R}) where {R<:Union{TropicalSemiringElem,MPolyRingElem{<:TropicalSemiringElem},PolyRingElem{<:TropicalSemiringElem}}}
-    return det(matrix(parent(first(A)),A))
+function det(A::Matrix{<:TropicalSemiringElem})
+  @req 0 < nrows(A) == ncols(A) "Non-square or empty matrix"
+  return det(matrix(parent(first(A)),A))
 end

test/TropicalGeometry/matrix.jl

@@ -5,9 +5,6 @@
         @testset "det(A::Generic.MatSpaceElem{<:TropicalSemiringElem})" begin
             A = Matrix(T[1 2; 3 5]) # (julia) matrix over tropical numbers
             @test det(A) == (minOrMax==min ? T(5) : T(6))
-            R, (x,y) = T[:x,:y]
-            A = Matrix(identity_matrix(R,2)) # (julia) matrix over tropical polynomials
-            @test det(A) == one(R)
         end
     end