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Some Equations Involving the Gamma Function

Sebastian Eterović, Adele Padgett

TL;DR

The paper proves that for any algebraic variety V ⊂ C^{2n} with no constant coordinates and with dim π_1(V) = n, the set of points in V lying on the graph of the Gamma function is Zariski dense in V. It develops a hybrid analytic–algebraic framework, leveraging Rouché's theorem and the Argument Principle, to control intersections of Gamma with algebraic systems, and extends the plane-curve n = 1 case to higher dimensions via a Rabinowitsch-trick reduction and a multivariable contour construction. Two proofs are given for the plane case: one via a general differential-transcendence result and another via explicit contour arguments yielding infinite solution sets and a lower bound on their distribution. The work situates Gamma within a broader existential closedness program, introduces Gamma-weakly special and Gamma-free notions, and provides corollaries on infinite periodic points, highlighting connections to Ax–Schanuel-type conjectures and potential generalizations to other transcendental functions.

Abstract

Let $V\subseteq\mathbb{C}^{2n}$ be an algebraic variety with no constant coordinates and with a dominant projection onto the first $n$ coordinates. We show that the intersection of $V$ with the graph of the $Γ$ function is Zariski dense in $V$.

Some Equations Involving the Gamma Function

TL;DR

The paper proves that for any algebraic variety V ⊂ C^{2n} with no constant coordinates and with dim π_1(V) = n, the set of points in V lying on the graph of the Gamma function is Zariski dense in V. It develops a hybrid analytic–algebraic framework, leveraging Rouché's theorem and the Argument Principle, to control intersections of Gamma with algebraic systems, and extends the plane-curve n = 1 case to higher dimensions via a Rabinowitsch-trick reduction and a multivariable contour construction. Two proofs are given for the plane case: one via a general differential-transcendence result and another via explicit contour arguments yielding infinite solution sets and a lower bound on their distribution. The work situates Gamma within a broader existential closedness program, introduces Gamma-weakly special and Gamma-free notions, and provides corollaries on infinite periodic points, highlighting connections to Ax–Schanuel-type conjectures and potential generalizations to other transcendental functions.

Abstract

Let be an algebraic variety with no constant coordinates and with a dominant projection onto the first coordinates. We show that the intersection of with the graph of the function is Zariski dense in .
Paper Structure (13 sections, 20 theorems, 60 equations, 2 figures)

This paper contains 13 sections, 20 theorems, 60 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notation and conventions
  3. Background on Gamma
  4. Behavior of Gamma in the upper right quadrant
  5. Transcendence conjectures for Gamma
  6. Geometric properties of Gamma
  7. The case of Plane Curves
  8. A general result about differentially transcendental functions
  9. The result for Gamma
  10. Adapting the method
  11. Proof of the Main Result
  12. The general set-up
  13. The proofs

Key Result

Theorem 1.1

Let $n$ be a positive integer and let $V\subseteq\mathop{\mathrm{\mathbb{C}}}\nolimits^{2n}$ be an algebraic variety with no constant coordinates. Let $\pi_1:\mathbb{C}^{2n}\to\mathbb{C}^n$ denote the projection onto the first $n$ coordinates. If $\dim \pi_1(V) = n$, then $V$ has a Zariski dense set

Figures (2)

  • Figure 1: Diagram of the curve $K(\beta,R)$.
  • Figure 2: Layout of $B(-A(\xi),r)$, $B(-A(\xi),2r)$, $U$ and $V$.

Theorems & Definitions (60)

  • Theorem 1.1
  • Lemma 2.1
  • Lemma 2.2
  • Corollary 2.3
  • Corollary 2.4
  • Proposition 2.5
  • proof
  • Lemma 2.6
  • proof
  • Lemma 2.7
  • ...and 50 more