Homaloidal determinants

TL;DR

This paper investigates which structured matrix determinants, especially Hankel matrices, are homaloidal polynomials, and explores their algebraic properties and inverse maps within polynomial rings.

Contribution

It identifies conditions under which determinants of structured matrices are homaloidal and analyzes their gradient ideals and algebraic invariants.

Findings

Hankel matrix determinants can be homaloidal under certain conditions

The gradient ideals exhibit specific primary decompositions and resolutions

Results on algebraic invariants of these determinants are established

Abstract

A form in a polynomial ring over a field is said to be homaloidal if its polar map is a Cremona map, i.e., if the rational map defined by the partial derivatives of the form has an inverse rational map. The object of this work is the search for homaloidal polynomials that are the determinants of sufficiently structured matrices. We focus on generic catatalecticants, with special emphasis on the Hankel matrix. An additional focus is on certain degenerations or specializations thereof. In addition to studying the homaloidal nature of these determinants, one establishes several results on the ideal theoretic invariants of the respective gradient ideals, such as primary components, multiplicity, reductions and free resolutions.