On the Determinantal Representation of Quaternary Forms

Abstract

We prove that a general polynomial form of degree d in 4 variables, over the complex field, can be written as the sum of two determinants of 2 × 2 matrices of forms, with given degree matrix (a _ij ), for any choice of non-negative integers a _ij  ≤ d with a ₁₁ + a ₂₂ = a ₁₂ + a ₂₁ = d.

Key Words:

• Join varieties
• Polynomial forms

2010 Mathematics Subject Classification:

• 14M12
• 13C40

Notes

Communicated by L. Ein.