Square root of the determinant of AB+I where A, B are skew-symmetric Imagine I have two skew-symmetric square matrices $A$, $B$. (So $A^\intercal = -A$, etc.) Now I am interested in the square root of the determinant of $AB+I$, where $I$ is the identity matrix, 
$$ x = \sqrt{ \det \left( AB + I \right) } $$
As quick inspection for small matrices suggests that this $x$ is a polynomial of the elements of $A$ and $B$, for example for $3 \times 3$ matrices we find
$$ x =  1 - a_{12} b_{12} - a_{13} b_{13} - a_{23} b_{23} $$
and I checked this analytically for matrices up to $6 \times 6$. It reminds me of the pfaffian of a skew-symmetric matrix, which is also a 'square root of a determinant' but nonetheless a polynomial in the matrix elements. 
Now my questions are:


*

*Does anyone know a proof that $x$ is a polynomial in the elements of $A$ and $B$, and if so, what is that polynomial?

*Does anyone know an efficient (so not $O(n!)$) algorithm to compute $x$?

 A: The quantity you are computing is very close to the relative Pfaffian, as found in for example Section 2 of this paper.
Let $A,B$ be skew matrices acting on $\mathbb{C}^{2n}$, then there is something called the relative Pfaffian of $A$ with respect to $B$, denoted $
 \operatorname{Pf}(A,B)$,
which satisfies the relation
\begin{equation}
 \operatorname{Pf}(A,B)^{2} = \det(I - AB).
\end{equation}
(So I guess you would be interested in $\operatorname{Pf}(-A,B)$.)
An explicit formula for the relative Pfaffian is given in Definition II.7 on page 8 (or 355 if you prefer). It reads
\begin{equation}
 \operatorname{Pf}(A,B) = \sum_{S} \operatorname{Pf}(A_{S}) \operatorname{Pf}(B_{S}),
\end{equation}
where the sum runs over non-empty even subsets $S$ of $\{1, ..., 2n\}$, and where $A_{S}$ denotes the restriction of $A$ to the subspace spanned by $\{e_{j}\}_{j \in S}$.
It is furthermore proven that if $A$ is invertible, then the relation
\begin{equation}
 \frac{\operatorname{Pf}(A^{-1} -B)}{\operatorname{Pf}(A^{-1})} = \operatorname{Pf}(A,B).
\end{equation}
holds, (Proposition II.6, page 7).
Just in case that link rots, the article is called "Pfaffians on Hilbert space" and is written by Arthur Jaffe, Andrzej Lesniewski, and Jonathan Weitsman, published in the journal of functional analysis.
