An improved upper bound for the Waring rank of the determinant

TL;DR

This paper presents a new, tighter upper bound for the Waring rank of the generic determinant, achieved through an explicit power sum decomposition and analysis of its symmetries.

Contribution

It introduces an improved upper bound for the Waring rank of the determinant using a novel explicit decomposition and symmetry analysis.

Findings

Upper bound for Waring rank is d * d!

Explicit power sum decomposition provided

Symmetries and defining equations of the decomposition analyzed

Abstract

The Waring rank of the generic $d \times d$ determinant is bounded above by $d \cdot d!$. This improves previous upper bounds, which were of the form an exponential times the factorial. Our upper bound comes from an explicit power sum decomposition. We describe some of the symmetries of the decomposition and set-theoretic defining equations for the terms of the decomposition.