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We compute the algebraic successive minima of the projective toric variety X_A associated to a finite set A \subset Z^n. As a consequence of this computation and the results of S.-W. Zhang on the distribution of small points, we derive estimates for the height of the variety X_A and of the A-resultant.
These estimates allow us to obtain an arithmetic analogue of the Kushnirenko's
theorem concerning the number of solutions of a polynomial equation system.
As an application of this result, we improve the previous estimates for
the height of the polynomials in the sparse Nullstellensatz.
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