[Paper Review] On the quantitative dynamical Mordell-Lang conjecture

Alina Ostafe, Min Sha|arXiv (Cornell University)|Jan 12, 2015

Algebraic Geometry and Number Theory16 references3 citations

TL;DR

This paper establishes the first quantitative bounds for the dynamical Mordell-Lang conjecture in algebraic dynamics, focusing on polynomial morphisms on affine space. It provides explicit upper bounds on the size of intersections between orbits and hypersurfaces, particularly for monomial systems with multiplicatively independent initial points, using results on exponential polynomials and linear forms in logarithms.

ABSTRACT

The dynamical Mordell-Lang conjecture concerns the structure of the intersection of an orbit in an algebraic dynamical system and an algebraic variety. In this paper, we bound the size of this intersection for various cases when it is finite.

Motivation & Objective

• To establish uniform quantitative bounds on the size of intersections between orbits of polynomial dynamical systems and algebraic hypersurfaces.
• To extend prior results on the dynamical Mordell-Lang conjecture beyond hyperplanes and uniform exponent maps to general monomial systems with distinct exponents.
• To provide explicit upper bounds in terms of the number of monomials in the defining polynomial of the hypersurface.
• To explore the applicability of methods based on finitely generated subgroups of (C*)^m and solutions to polynomial-exponential equations.
• To investigate the possibility of extending results to non-monomial systems via conjugation by polynomial automorphisms.

Proposed method

• Use of estimates for integer solutions of polynomial-exponential equations derived from Schlickewei and Schmidt's results on linear forms in logarithms.
• Leverage uniform bounds on the number of zeros of non-degenerate linear recurrence sequences from [1, 2, 19] for power map cases.
• Apply Corollary 2.7 to bound solutions in finitely generated subgroups of (C*)^m, requiring that all iterates Φ^(n)(w) lie in a single such group.
• Transform the orbit intersection problem into a system of exponential Diophantine equations via monomial dynamics.
• Use multiplicative independence of coordinates to ensure the structure of solutions is constrained within a finitely generated group.
• Apply the theory of exponential polynomials and their zero sets to bound the number of solutions to G(Φ^(n)(w)) = 0.

Experimental results

Research questions

• RQ1 What is the maximum number of times an orbit under a monomial morphism can intersect a hypersurface in affine space?
• RQ2 Can uniform upper bounds be established for the size of the intersection set S_w(Φ,V) across different initial points and hypersurfaces?
• RQ3 How does the number of monomials in the defining polynomial G of the hypersurface affect the size of the intersection?
• RQ4 To what extent can the methods based on finitely generated subgroups of (C*)^m be extended to non-monomial dynamical systems?
• RQ5 Can the techniques be adapted to bound the synchronized intersection of two distinct orbits in the same space?

Key findings

• For a diagonal monomial map Φ = (X₁^d, ..., Xₘ^d) with d ≥ 2, and any hypersurface V defined by a polynomial G with 𝔫(G) monomials, the intersection size |S_w(Φ,V)| is at most (8𝔫(G))^{4𝔫(G)^5} when w has multiplicatively independent coordinates.
• The bound is uniform in the sense that it depends only on the number of monomials in G, not on the specific coefficients or degrees.
• The result extends Silverman and Viray’s work on the power map and hyperplanes to general hypersurfaces and distinct exponents in monomial systems.
• The method relies on the orbit lying in a fixed finitely generated subgroup of (C*)^m, which holds for monomial maps but not in general.
• The bound can be improved in special cases, such as when the absolute values of the coordinates of w are multiplicatively independent and one monomial dominates in modulus.
• The approach can be adapted to bound the synchronized intersection of two orbits by embedding the problem into a higher-dimensional system and applying the same techniques.