Waring problems and the Lefschetz properties

TL;DR

This paper explores how Lefschetz properties influence various ranks in polynomial Waring problems, introduces the mixed Hessian matrix as a key tool, and constructs new wild forms with unique rank characteristics.

Contribution

It connects Lefschetz properties with Waring ranks and introduces new families of wild forms exhibiting higher cactus rank than border rank.

Findings

Lefschetz properties impact Waring, border, and cactus ranks

Introduction of mixed Hessian matrix as a main analytical tool

Construction of new wild forms with distinct rank behaviors

Abstract

We study three variations of the Waring problem for polynomials, concerning the Waring rank, the border rank and the cactus rank of a form and we show how the Lefschetz properties of the associated algebra affect them. The main tool is the theory of mixed Hessian matrix. We construct new families of wild forms, that is, forms whose cactus rank, of schematic nature, is bigger then the border rank, defined geometrically.