This material has been published in
Algebra Appl. 426 (2007), 159-189,
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2-adic valuations of certain ratios of products of
factorials and applications
We prove the conjecture of Falikman-Friedland-Loewy
on the parity of the degrees of projective varieties of
nxn complex symmetric matrices of rank at most k.
We also characterize the parity of the degrees of projective
varieties of nx
n complex skew symmetric matrices of rank at most 2p.
We give recursive relations which determine the parity of the
degrees of projective varieties of mxn complex matrices
of rank at most k.
In the case the degrees of these varieties are odd, we characterize
the minimal dimensions of subspaces of nx
n skew symmetric real matrices and of mxn
real matrices containing a nonzero matrix of rank at most k.
The parity questions studied here are also of combinatorial interest
since they concern the parity of the number of plane partitions
contained in a given box, on the one hand, and the parity of
the number of symplectic tableaux of rectangular shape, on the
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