An eventual jet test for Laurent polynomials on a triangle

Proved

OAI.MaximalSeshadri.Interpolation.eventual_torus_jet_test

by wurtle · Nov 4, 2026 · Mathlib 0df444a (Lean v4.33.1)

maximal-seshadrioai-math

Let $d,H>0$ , $T>1$ . There is $r_0>0$ such that for $r\ge r_0$ there are $t>T$ and $r$ distinct points of $(\mathbb C^*)^2$ with the following property. If $k,m>0$ with $k\sqrt{H/r}<m$ , and $s$ is a nonempty finite set of exponents in the scaled compressedTriangle $k\cdot\operatorname{conv}\{(-dH/(1+t),0),(dHt/(1+t),0),(0,1/d)\}$ with nonzero coefficients $c$ , then the Laurent polynomial with exponents $s$ and coefficients $c$ has a mixed $m$ -jet which is nonzero at some point.

Formalization Note The statement is verbatim from OpenAI's Lean development, with Definitions files of this development imported in place of the upstream modules; these are supporting code for the statement.

Preamble

import Mathlib
import Definitions.Def_MaximalSeshadriConstants
import Definitions.Def_SeshadriPrelim6

namespace OAI

section
namespace MaximalSeshadri.Interpolation
open scoped BigOperators Pointwise

Formal statement

theorem eventual_torus_jet_test (d H T : ℝ) (hd : 0 < d) (hH : 0 < H) (hT : 1 < T) :
    ∃ r₀ : ℕ, 0 < r₀ ∧ ∀ r : ℕ, r₀ ≤ r →
      ∃ t : ℝ, T < t ∧ ∃ points : Fin r → ℂ × ℂ,
        Function.Injective points ∧ (∀ l, (points l).1 ≠ 0 ∧ (points l).2 ≠ 0) ∧
        ∀ (k m : ℕ), 0 < k → 0 < m → (k : ℝ) * Real.sqrt (H / r) < (m : ℝ) →
          ∀ (s : Finset (ℤ × ℤ)), s.Nonempty →
            (∀ p ∈ s, ((p.1 : ℝ), (p.2 : ℝ)) ∈ (k : ℝ) • compressedTriangle d H t) →
            ∀ c : ℤ × ℤ → ℂ, (∀ p ∈ s, c p ≠ 0) →
              ∃ l : Fin r, ¬ MixedJetZero (laurentEval s c) (points l).1 (points l).2 m := by
  sorry

end MaximalSeshadri.Interpolation
end
end OAI

Source

OpenAI, Maximal Seshadri constants on arbitrary polarized surfaces, September 23, 2026, https://github.com/openai/math (paper: preprints/Maximal-Seshadri-Constants-on-Arbitrary-Polarized-Surfaces-September-23-2026/main.pdf; Lean: lean/OAI/AlgebraicGeometry/Seshadri, Apache License 2.0); Lean declaration `OAI.MaximalSeshadri.Interpolation.eventual_torus_jet_test`, https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/AlgebraicGeometry/Seshadri/Interpolation/TorusJets.lean#L158

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