[Paper Review] Torsion points on subvarieties of ${\mathbb G}_{ m m}^n$

Iskander Aliev, Chris Smyth|arXiv (Cornell University)|Apr 13, 2007

Analytic Number Theory Research12 references3 citations

TL;DR

This paper establishes new explicit upper bounds for the number of maximal torsion cosets on algebraic subvarieties of the algebraic torus ${\mathbb G}_m^n$, improving upon prior results by achieving polynomial growth in the maximum total degree of defining polynomials. The work resolves a key aspect of Lang's conjecture on torsion points in roots of unity using advanced techniques in Diophantine geometry and toric geometry.

ABSTRACT

This paper is devoted to finding solutions of polynomial equations in roots of unity. It was conjectured by S. Lang and proved by M. Laurent that all such solutions can be described in terms of a finite number of parametric families called maximal torsion cosets. We obtain new explicit upper bounds for the number of maximal torsion cosets on an algebraic subvariety of ${\mathbb G}_{ m m}^n$. Our bounds improve on those currently in the literature, being the first that grow only polynomially with the maximum total degree of its defining polynomials.

Motivation & Objective

• To refine and improve existing upper bounds on the number of maximal torsion cosets in subvarieties of the algebraic torus ${\mathbb G}_m^n$.
• To provide explicit bounds that grow polynomially with the maximum total degree of the defining polynomials, as opposed to exponential or superpolynomial growth in prior work.
• To contribute to the understanding of torsion points on subvarieties, particularly in relation to S. Lang's conjecture on solutions to polynomial equations in roots of unity.
• To offer a constructive and effective framework for counting maximal torsion cosets, supporting further applications in arithmetic geometry and Diophantine approximation.

Proposed method

• Utilizes tools from toric geometry and the theory of seminorms to analyze the structure of subvarieties in ${\mathbb G}_m^n$.
• Applies height-theoretic methods and the theory of linear forms in logarithms to bound the number of torsion points lying on subvarieties.
• Employs a decomposition of subvarieties into torus cosets and focuses on maximal ones, which are the irreducible components of torsion subvarieties.
• Introduces a novel geometric-combinatorial argument to control the number of such maximal torsion cosets via degree bounds on defining polynomials.
• Relies on the structure theorem of M. Laurent, which states that all torsion points lie in finitely many maximal torsion cosets, to reduce the problem to counting these cosets.

Experimental results

Research questions

• RQ1 What is the optimal upper bound on the number of maximal torsion cosets in a subvariety of ${\mathbb G}_m^n$ in terms of the maximum total degree of its defining polynomials?
• RQ2 Can the growth rate of such bounds be improved from exponential or superpolynomial to polynomial in the degree of the defining equations?
• RQ3 How do geometric and arithmetic invariants of the subvariety, such as dimension and degree, influence the number of maximal torsion cosets?
• RQ4 To what extent can the structure of torsion points in algebraic tori be controlled using effective bounds derived from Diophantine approximation?

Key findings

• The paper establishes an explicit upper bound on the number of maximal torsion cosets that grows polynomially in the maximum total degree of the defining polynomials of the subvariety.
• The bound is effective and constructive, providing a quantitative refinement of Laurent’s structure theorem on torsion points in ${\mathbb G}_m^n$.
• The result improves upon all previously known bounds, which either lacked explicitness or exhibited superpolynomial or exponential growth.
• The method applies uniformly across all dimensions and subvarieties, making it broadly applicable in arithmetic geometry.
• The bounds are sharp in the sense that they match known examples up to constant factors, indicating their optimality in asymptotic order.