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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2603.19184 |
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| _version_ | 1866915960405884928 |
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| author | Hoşten, Serkan Kurylenko, Vadym Neuhaus, Elke Rieke, Nikolas |
| author_facet | Hoşten, Serkan Kurylenko, Vadym Neuhaus, Elke Rieke, Nikolas |
| contents | 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 $Δ_1 \times Δ_1 \times Δ_n$ 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 $Δ_1 \times Δ_1 \times Δ_1$ this Euler characteristic depends only on the vanishing patterns of the factors of the principal $A$-determinant, but this fails for $Δ_1 \times Δ_1 \times Δ_n$ 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 $\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_19184 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Euler Stratification for $\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^n$ Hoşten, Serkan Kurylenko, Vadym Neuhaus, Elke Rieke, Nikolas Algebraic Geometry Commutative Algebra Combinatorics 14F45, 13P15, 05E14, 62R01 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 $Δ_1 \times Δ_1 \times Δ_n$ 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 $Δ_1 \times Δ_1 \times Δ_1$ this Euler characteristic depends only on the vanishing patterns of the factors of the principal $A$-determinant, but this fails for $Δ_1 \times Δ_1 \times Δ_n$ 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 $\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^2$. |
| title | The Euler Stratification for $\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^n$ |
| topic | Algebraic Geometry Commutative Algebra Combinatorics 14F45, 13P15, 05E14, 62R01 |
| url | https://arxiv.org/abs/2603.19184 |