Successive minima of projective toric varieties

2002/09/16 by Sombra, Martin
#14M25 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT) #Primary: 11G50 #Secondary: 14G40

paper · doi:10.48550/arxiv.math/0209195

Abstract

We compute the successive minima of the projective toric variety X_\cA associated to a finite set \cA ⊂ \Zⁿ. As a consequence of this computation and of the results of S.-W. Zhang on the distribution of small points, we derive estimates for the height of the subvariety X_\cA and of the \cA-resultant. These estimates allow us to obtain an arithmetic analogue of the Bezout-Kushnirenko's theorem concerning the number of solutions of a system of polynomial equations. As an application of this result, we improve the known estimates for the height of the polynomials in the sparse Nullstellensatz.

Related

• Successive minima of toric height functions
• Diophantine geometry and toric varieties
• Notes on toric varieties from Mori theoretic viewpoint
• Secant varieties and successive minima
• Approximation of adelic divisors and equidistribution of small points