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PreprintCC BY 4.0arXiv

The Logarithmic A-Discriminant

  • Madeleine Goertz
  • Bernd Sturmfels
  • Simon Telen
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Abstract

We extend the GKZ theory of discriminants to the setting of Euler integrals where both coefficients and exponents are variables. The integrand is the likelihood function of a toric model in algebraic statistics, whose number of critical points is the volume of the Newton polytope. Our logarithmic A-discriminant vanishes when two critical points come together. This discriminant is a hypersurface, parametrized by the logarithmic Horn uniformization. Hurwitz forms of toric varieties yield a degree formula, and a description of leading coefficients. We emphasize applications and computations.

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Cite

BibTeXRIS
BibTeX
@misc{goertz2026logarithmic,
  title = {{The Logarithmic A-Discriminant}},
  author = {Madeleine Goertz and Bernd Sturmfels and Simon Telen},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.29204},
  eprint = {2610.29204},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.29204},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - The Logarithmic A-Discriminant
AU  - Madeleine Goertz
AU  - Bernd Sturmfels
AU  - Simon Telen
PY  - 2026
DA  - 2026/10/21
PB  - arXiv
DO  - 10.48550/arxiv.2610.29204
UR  - https://arxiv.org/abs/2610.29204
AB  - We extend the GKZ theory of discriminants to the setting of Euler integrals where both coefficients and exponents are variables. The integrand is the likelihood function of a toric model in algebraic statistics, whose number of critical points is the volume of the Newton polytope. Our logarithmic A-discriminant vanishes when two critical points come together. This discriminant is a hypersurface, parametrized by the logarithmic Horn uniformization. Hurwitz forms of toric varieties yield a degree formula, and a description of leading coefficients. We emphasize applications and computations.
ER  -