Bounds in polynomial rings over Artinian local rings

Let R be a (mixed characteristic) Artinian local ring of length l and let X be an n-tuple of variables. We prove that several algebraic constructions in the ring R[X] admit uniform bounds on the degrees of their output in terms of l, n and the degrees of the input. For instance, if I is an ideal in R[X] generated by polynomials g(i) of degree at most d and if f is a polynomial of degree at most d belonging to I, then f = q(1)f(1) + center dot center dot center dot + q(s)f (s) , for some q(i) of degree bounded in terms of d, l and n only. Similarly, the module of syzygies of I is generated by tuples all of whose entries have degree bounded in terms of d, l and n only.