The number of maximal torsion cosets in subvarieties of tori

2015/09/19 by César Martínez, Martínez, César
Mathematics · #FOS: Mathematics #Number Theory (math.NT) #Primary 11G35 #Secondary 14G25 #math.NT #msc:11G35 #msc:14G25

paper · pdf · doi:10.48550/arxiv.1509.05898

21 pages

arxiv created 2015/09/19 · arxiv updated 2015/09/22

Abstract

We present sharp bounds on the number of maximal torsion cosets in a subvariety of the complex algebraic torus \mathbbG_\textrmmⁿ. 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.

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