Primary Decomposition of Lattice Basis Ideals

We study the primary decomposition of lattice basis ideals. These ideals are binomial ideals with generators given by the elements of a basis of a saturated integer lattice. We show that the minimal primes of such an ideal are completely determined by the sign pattern of the basis elements, while the embedded primes are not. As a special case we examine the ideal generated by the 2 × 2 adjacent minors of a generic m × n matrix. In particular, we determine all minimal primes in the 3 × n case. We also present faster ways of computing a generating set for the associated toric ideal from a lattice basis ideal.