Generalized Einstein Laurent polynomials, Toric Kähler-Einstein Rigidity, and Finite Exponential Families
Abstract
We study when the logarithm ψ of a positive finite exponential sum on ℝd satisfies det∇2ψ=Cexp(⟨ b,θ⟩−λψ). This is the Kähler--Einstein equation for metrics induced by exponential maps into projective space and, for natural exponential families, the condition that the Jeffreys prior be a Diaconis--Ylvisaker conjugate prior. First, we classify the bivariate Laurent polynomials with unimodular support satisfying the generalized Einstein condition of Di Scala and Sombra: up to units and monomial changes of coordinates, they are powers of an affine trinomial or products of powers of two independent binomials. Second, we show in every dimension that a smooth compact toric manifold with a projectively induced Kähler--Einstein metric is a product of projective spaces with matched multiples of the Fubini--Study metrics, immersed by a complete Veronese--Segre system up to automorphisms. This proves the compact toric case of the homogeneity conjecture for such metrics and the fixed-point germ and univalent forms of a conjecture of Manno and Salis. Third, without lattice or rationality assumptions, the finite-support exponential families satisfying the equation are, up to affine changes of statistic, exactly the products of multinomial families with a common ratio of categories to trials; this settles the finite-support case of a question of Casalis.
Related
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BibTeX
@misc{huang2026generalized,
title = {{Generalized Einstein Laurent polynomials, Toric Kähler-Einstein Rigidity, and Finite Exponential Families}},
author = {Shaosai Huang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.18067},
eprint = {2609.18067},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.18067},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Generalized Einstein Laurent polynomials, Toric Kähler-Einstein Rigidity, and Finite Exponential Families
AU - Shaosai Huang
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.18067
UR - https://arxiv.org/abs/2609.18067
AB - We study when the logarithm ψ of a positive finite exponential sum on ℝ^d satisfies det∇^2ψ=Cexp(⟨ b,θ⟩−λψ). This is the Kähler--Einstein equation for metrics induced by exponential maps into projective space and, for natural exponential families, the condition that the Jeffreys prior be a Diaconis--Ylvisaker conjugate prior. First, we classify the bivariate Laurent polynomials with unimodular support satisfying the generalized Einstein condition of Di Scala and Sombra: up to units and monomial changes of coordinates, they are powers of an affine trinomial or products of powers of two independent binomials. Second, we show in every dimension that a smooth compact toric manifold with a projectively induced Kähler--Einstein metric is a product of projective spaces with matched multiples of the Fubini--Study metrics, immersed by a complete Veronese--Segre system up to automorphisms. This proves the compact toric case of the homogeneity conjecture for such metrics and the fixed-point germ and univalent forms of a conjecture of Manno and Salis. Third, without lattice or rationality assumptions, the finite-support exponential families satisfying the equation are, up to affine changes of statistic, exactly the products of multinomial families with a common ratio of categories to trials; this settles the finite-support case of a question of Casalis.
ER -