Successive minima of projective toric varieties

We compute the successive minima of the projective toric variety $X_{\text{cA}}$ associated to a finite set $\text{cA}\subset {\mathbb{Z}}^n$. 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 $X_{\text{cA}}$ and of the $\text{cA}$-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.