Expressing Forms as a sum of Pfaffians

TL;DR

This paper investigates the minimal number of Pfaffian sums needed to express a general form in four variables, establishing an upper bound related to the matrix's size for homogeneous symmetric matrices.

Contribution

It introduces a bound on the number of Pfaffians required to represent forms, specifically proving s(A) < k+1 for homogeneous matrices in four variables.

Findings

Established an upper bound s(A) < k+1 for all homogeneous matrices A in four variables.

Connected the degree of forms with the trace of the matrix A.

Provided a new approach to expressing forms as sums of Pfaffians.

Abstract

Let A= (a_{ij}) be a symmetric non-negative integer 2k x 2k matrix. A is homogeneous if a_{ij} + a_{kl}=a_{il} + a_{kj} for any choice of the four indexes. Let A be a homogeneous matrix and let F be a general form in C[x_1, \dots x_n] with 2deg(F) = trace(A). We look for the least integer, s(A), so that F= pfaff(M_1) + \cdots + pfaff(M_{s(A)}), where the M_i's 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+1 for all A.