The Euler Stratification for $\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^n$

TL;DR

This paper investigates the Euler characteristic of hypersurfaces in a product of complex tori, linking it to algebraic statistics models and exploring its dependence on polynomial support and stratification.

Contribution

It establishes the relationship between Euler characteristic and ML degree in three-way independence models, and characterizes the Euler stratification for specific cases.

Findings

Euler characteristic equals ML degree for the hypersurfaces studied.

Dependence of Euler characteristic on vanishing patterns varies with dimension.

All positive integers up to the maximum ML degree are realizable as Euler characteristics.

Abstract

We study the Euler characteristic of a hypersurface in $(\mathbb{C}^*)^2 \times (\mathbb{C}^*)^n$ defined by a polynomial whose monomial support corresponds to lattice points in $\Delta_1 \times \Delta_1 \times \Delta_n$ as the coefficients of the defining polynomial vary. Each member of this hypersurface family corresponds to a three-way independence model from algebraic statistics, and the (signed) Euler characteristic is equal to the maximum likelihood degree (ML degree) of the model. We show in the case of $\Delta_1 \times \Delta_1 \times \Delta_1$ this Euler characteristic depends only on the vanishing patterns of the factors of the principal $A$-determinant, but this fails for $\Delta_1 \times \Delta_1 \times \Delta_n$ with $n \geq 2$. We prove that, for all $n\geq 1$, all positive integers up to the maximum possible ML degree can be realized as the Euler characteristic. Furthermore, we completely determine the Euler stratification for $\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1$ and provide partial information for $\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^2$.