Grobner bases of toric ideals

We study here \grobner\ bases of ideals which define toric varieties. We connect these ideals with the sub-lattices of $Z^d$, then deduce properties on their \grobner\ bases, and give applications of these results. The main contributions of the report are a bound on the degree of the \grobner\ bases, the fact that they contain Minkowski successive minima of a lattice (in particular shortest vector), and the algorithm (derived from Buchberger algorithm), which starts with ideal of polynomials with less variables than usual