Persistent transcendental B\'ezout theorems

TL;DR

This paper demonstrates that a modified coarse count, inspired by topological data analysis, can effectively bound the zeros of holomorphic self-mappings in higher dimensions, countering classical predictions.

Contribution

It introduces a new bound for zeros of holomorphic maps using persistence modules, extending classical results with a novel topological approach.

Findings

Counterexample to classical zero count predictions in higher dimensions

Introduction of a persistence module-inspired bound for holomorphic maps

Extension of Bezout theorems using topological data analysis techniques

Abstract

An example of Cornalba and Shiffman from 1972 disproves in dimension two or higher a classical prediction that the count of zeros of holomorphic self-mappings of the complex linear space should be controlled by the maximum modulus function. We prove that such a bound holds for a modified coarse count inspired by the theory of persistence modules originating in topological data analysis.