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A Probabilistic Proof of a Positivstellensatz of Baldi-Sinn-Telek-Weigert

Primarily AI-generated textHuman understanding: no partsmath.AG — Algebraic Geometrymath.PR — Probability

Contributed by Colin Tan ↗

Version 1 / Oct 16, 2026 / CC0 1.0

Abstract

The AI gives a probabilistic proof of a uniform-denominator Positivstellensatz of Baldi--Sinn--Telek--Weigert for Laurent polynomials whose induced homogeneous form is strictly positive on the nonnegative part of a projective toric variety. Only this abstract was human-written. Three classical tools from mathematical statistics and probability theory were used by the AI: the Barndorff-Nielsen completion of discrete exponential families, sufficiency-based invariance under exponential tilting following Halmos--Savage and Van Campenhout--Cover, and the conditional limit theorem (equivalence of ensembles) of Diaconis--Freedman.

Provenance statement

The classical Positivstellensatz of Polya over the simplex is known to be a consequence of the approximation of the Multivariate Hypergeometric Distribution by the Multinomial Distribution, as already known at least implicitly by de Klerk-Laurent-Sun. Recently, Baldi-Sinn-Telek-Weigert generalised Polya's Positivstellensatz to a Positivstellensatz where the uniform denominator could be any Laurent polynomial whose coefficients are all 1. Since Polya's Positivstellensatz has this proababilistic proof, there ought to be an analogous probabilistic proof of the Baldi-Sinn-Telek-Weigert Positivstellensatz. This belief led me to prompt Gemini Argon 4 for such a proof. I changed the title and the abstract, but left Gemini to speak for herself in the rest of the paper.

Tools used

Google
GeminiVersion Argon 4

References

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Version history

  1. v1Submitted by Colin TanInitial depositCurrentOct 16, 2026