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We present a new approach to the ideal membership problem for polynomial rings over the integers: given polynomials $f_0,f_1,...,f_n\in\Z[X]$, where $X=(X_1,...,X_N)$ is an $N$-tuple of indeterminates, are there $g_1,...,g_n\in\Z[X]$ such that $f_0=g_1f_1+...+g_nf_n$? We show that the degree of the polynomials $g_1,...,g_n$ can be bounded by $(2d)^{2^{O(N^2)}}(h+1)$ where $d$ is the maximum total degree and $h$ the maximum height of the coefficients of $f_0,...,f_n$. Some related questions, primarily concerning linear equations in $R[X]$, where $R$ is the ring of integers of a number field, are also treated., Comment: 34 pages

We prove that for any fixed d the generating function of the projection of the set of integer points in a rational d-dimensional polytope can be computed in polynomial time. As a corollary, we deduce that various interesting sets of lattice points, notably integer semigroups and (minimal) Hilbert bases of rational cones, have short rational generating functions provided certain parameters (the dimension and the number of generators) are fixed. It follows then that many computational problems for such sets (for example, finding the number of positive integers not representable as a non-negative integer combination of given coprime positive integers a_1 ... a_d admit polynomial time algorithms. We also discuss a related problem of computing the Hilbert series of a ring generated by monomials., 26 pages