Values of permanent and positive solution of Wang-Krauter problem

The talk is based on the joint work with M.V. Budrevich.
The class of $(-1,1)$-matrices is very important in algebra and combinatorics and in various their applications. For example, well-known Hadamard matrices are of this type.
An important matrix function is the permanent:
$$ {\rm per}\, A= \sum_{\sigma\in { S}_n} a_{1\sigma(1)}\cdots a_{n\sigma(n)}, $$
here $A=(a_{ij})\in M_n({\mathbb F})$ is an $n\times n$ matrix over a field ${\mathbb F}$ and $S_n$ denotes the set of all permutations of the set $\{1,\ldots, n\}$.
While the computation of the determinant can be done in a polynomial time, it is still an open question, if there are such algorithms to compute the permanent.
In this talk we discuss the permanents of $\pm 1$-matrices.
In 1974 Wang posed a problem to find a decent upper bound for $|{\rm per}(A)|$ if $A$ is a square $\pm 1$-matrix of rank $k$. In 1985 Krauter conjectured some concrete upper bound.
We prove the Krauter's conjecture and thus obtain the complete answer to the Wang's question. In particular, we characterized matrices with the maximal possible permanent for each value of $k$.