Let $I_1\subset I_2\subset\dots$ be an increasing sequence of ideals of the ring $\Bbb Z[X]$, $X=(x_1,\dots,x_n)$ 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 $\Bbb Q I$ of the ring $\Bbb Q[X]$ and the the ideals $I\otimes(\Bbb Z/m\Bbb Z)$ of the rings $(\Bbb Z/m\Bbb Z)[X]$ are known. Such an algorithmic problem arises, for example, in the construction of Markov and semi-Markov traces on cubic Hecke algebras.

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