A rational-function derivative cannot be a nonzero simple pole

Proved

OAI.PiExponent.ratFunc_derivation_ne_simple_pole

by marwahaha · Oct 7, 2026 · Mathlib 0df444a (Lean v4.33.1)

differential-algebranumber-theoryopenai-pi-exponent-017

Let $K$ be a field of characteristic zero, and let $D$ be a $K$ -derivation of $K(X)$ satisfying $D(X)=1$ . For every $r\in K(X)$ and every nonzero $c\in K$ ,

D(r)\ne c/X.

This obstructs rational antiderivatives of a nonzero logarithmic differential.

Preamble

import Mathlib.Algebra.Polynomial.Derivation
import Mathlib.Algebra.Polynomial.Div
import Mathlib.RingTheory.LaurentSeries
import Mathlib.RingTheory.Trace.Defs
import Mathlib.Tactic


namespace OAI
namespace PiExponent
end PiExponent
end OAI
open scoped LaurentSeries Matrix
open Polynomial
open OAI.PiExponent

Formal statement

theorem OAI.PiExponent.ratFunc_derivation_ne_simple_pole {K : Type*} [Field K] [CharZero K]
    (D : Derivation K (RatFunc K) (RatFunc K))
    (hDX : D RatFunc.X = 1) (r : RatFunc K) {c : K} (hc : c ≠ 0) :
    D r ≠ RatFunc.C c / RatFunc.X := by sorry

Source

https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/NumberTheory/PiExponent/Analysis/LogarithmicObstruction.lean#L75-L99

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