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 $n\times n$ complex symmetric matrices of rank at most $k$.   We also characterize the parity of the degrees of projective varieties of $n\times n$ complex skew symmetric matrices of rank at most $2p$.   We give recursive relations which determine the parity of the degrees of projective varieties of $m\times n$ 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 $n\times n$ skew symmetric real matrices and of $m\times n$ 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 other hand. 
Shmuel Friedland (University of Illinois at Chicago) and Christian   Krattenthaler (Universit\'e Claude Bernard Lyon-I)