The number of maximal torsion cosets in subvarieties of tori

TL;DR

This paper establishes precise bounds on the number of maximal torsion cosets in subvarieties of complex algebraic tori, confirming conjectures relating to torsion points and geometric properties of hypersurfaces.

Contribution

It provides new bounds based on polynomial degree and toric degree, and proves conjectures on torsion points in hypersurfaces.

Findings

Bounds in terms of polynomial degree and toric degree

Proof of Ruppert and Aliev-Smyth conjectures

Bounds relate to Newton polytope volume and multidegree

Abstract

We present sharp bounds on the number of maximal torsion cosets in a subvariety of the complex algebraic torus $\mathbb{G}_{\textrm{m}}^n$. Our first main result gives a bound in terms of the degree of the defining polynomials. A second result gives a bound in terms of the toric degree of the subvariety. As a consequence, we prove the conjectures of Ruppert and of Aliev and Smyth on the number of isolated torsion points of a hypersurface. These conjectures bound this number in terms of the multidegree and the volume of the Newton polytope of a polynomial defining the hypersurface, respectively.