On the Modular Computation of Gröbner Bases with Integer Coefficients

Let I1 ⊂ I2 ⊂ . . . be an increasing sequence of ideals of the ring Z[X], X = (x1, . . . , xn), and let I be their union. We propose an algorithm to compute the Gr¨obner base of I under the assumption that the Gr¨obner bases of the ideal QI of the ring Q[X] and of the ideals I ⊗ (Z/mZ) of the rings (Z/mZ)[X] are known. Such an algorithmic problem arises, for example, in the construction of Markov and semi-Markov traces on cubic Hecke algebras.