The number of maximal torsion cosets in subvarieties of tori

Abstract

Abstract We present sharp bounds on the number of maximal torsion cosets in a subvariety of the complex algebraic torus 𝔾 m n {\mathbb{G}_{\mathrm{m}}^{n}} . Our first main result gives a bound in terms of the degree of the defining polynomials. We also give a bound for the number of isolated torsion point, that is maximal torsion cosets of dimension 0, in terms of the volume of the Newton polytope of the defining polynomials. This result proves 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.