parallel algorithms for the determinant of the Hessenberg matrix I am interested in highly parallel algorithms for computing the determinant of matrices of a special form (over finite fields).

It is known that computing determinant of general matrices over finite fields is in NC2.
I've been wondering whether something better is known (hopefully NC1) for the (say, lower) Hessenberg matrix.

If it helps, it is further restricted so that its subdiagonal is all 1's, and all I care about is whether the determinant is 0 or not.
Sorry, this is clearly unlikely.
Otherwise, it would be possible to use the algorithm to solve any NL\poly
problem in NC1 via a reduction to evaluating the determinant of a Hessenberg matrix over a sufficiently large field.
The reduction appears as Lemma 3 (and related discussion) in
"Perfect constant-round secure computation via perfect randomizing polynomials".