Expressing Forms as a Sum of Pfaffians

Abstract

Let A = (aij) be a symmetric non-negative integer 2k X 2k matrix. A is homogeneous if aij + ail = an + akj for any choice of the four indexes. Let A be a homogeneous matrix and let F be a general form in C[xi,....xn] with 2deg(F) = trace(A). We look for the least integer s(A), so that F = pfaff(M1) + ••• + pfaff(Ms(A)) where the Mi = (Fim) are 2k X 2k skew-symmetric matrices of forms with degree matrix A. We consider this problem for n = 4 and we prove that s (A)_< k for all A.