Expressing a General Form as a Sum of Determinants

TL;DR

This paper investigates the minimal number of determinant sums needed to express a form with a given degree matrix, establishing bounds and conditions based on matrix properties.

Contribution

It introduces the concept of s(A) for homogeneous matrices and provides bounds and exact values for s(A) depending on matrix size and entries.

Findings

s(A) is at most k^{n-3} for n>3

s(A) < k^{n-3} in infinitely many cases

s(A) equals k^{n-3} when matrix entries are large

Abstract

Let A= (a_{ij}) be a non-negative integer k x k matrix. A is a homogeneous matrix if a_{ij} + a_{kl}=a_{il} + a_{kj} for any choice of the four indexes. We ask: If A is a homogeneous matrix and if F is a form in C[x_1, \dots x_n] with deg(F) = trace(A), what is the least integer, s(A), so that F = det M_1 + ... + det M_{s(A)}, where the M_i's are k x k matrices of forms with degree matrix A? We consider this problem for n>3 and we prove that s(A) is at most k^{n-3} and s(A) <k^{n-3} in infinitely many cases. However s(A) = k^{n-3} when the entries of A are large with respect to k.