On the structure and generic non-Cartesianity of polynomials in product spaces

TL;DR

This paper develops a comprehensive theory of Cartesian and non-Cartesian polynomials on product complex spaces, showing non-Cartesian polynomials are generic and providing criteria and algorithms for their identification.

Contribution

It introduces a general framework for understanding polynomial structure in product spaces, including criteria and algorithms for non-Cartesianity, and connects these concepts to incidence geometry.

Findings

Generic polynomials are non-Cartesian in broad dimensions

Effective criteria for non-Cartesianity are established

Algorithms using Gröbner bases decide Cartesian structure

Abstract

We develop a general theory of Cartesian and non-Cartesian polynomials on products of complex spaces $\mathbb{C}^{n_1} \times \cdots \times \mathbb{C}^{n_k}$. We prove that, for any fixed degree $d \ge 2$, a (Zariski) generic polynomial is non-Cartesian in a broad range of dimensions, establishing that Cartesian structure is highly exceptional. We further introduce effective sufficient criteria for a polynomial to be non-Cartesian. Moreover, we show that being (non)-Catersian can be decided algorithmically via Gr\"obner basis methods and quantitative forms of Hilbert's Nullstellensatz. As an application, we connect the non-Cartesian condition to incidence geometry, obtaining sharp intersection bounds and constructing extremal configurations that demonstrate the optimality of these estimates.