On the Koiran-Skomra's question about Hessians

TL;DR

This paper provides a negative answer to a question about Hessians posed by Koiran and Skomra, motivated by Kayal’s algorithm, and conjectures that the counterexamples are unique, also exploring a local variant of the question.

Contribution

It offers the first negative resolution to Koiran and Skomra’s Hessian question and introduces a conjecture about the uniqueness of the counterexamples.

Findings

Counterexamples to Koiran and Skomra’s Hessian question are identified.

Conjecture that these counterexamples are the only ones.

Analysis of a local version of the Hessian question.

Abstract

We give a negative answer to a question of Koiran and Skomra about Hessians, motivated by Kayal's algorithm for the equivalence problem to the Fermat polynomial. We conjecture that our counterexamples are the only ones. We also study a local version of their question.