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For an ideal I in a regular local ring (R,m)$ with residue class field K =R/m or a standard graded K-algebra R we show that for k >> 0  --> the Artin--Rees number of the syzygy modules of I^k as submodules of thefree modules from a free resolution is constant, and thereby present theArtin-Rees number as a proper replacement of regularity in the local situation,  --> the ring R/I^k is Golod, its Poincer{\'e}-Betti series is rational andthe Betti numbers of the free resolution of K over R/I^k are polynomials in kof a specific degree.  The first result is an extension of work of Kodiyalam and Cutkosky, Herzog &Trung on the regularity of I^k for k >> 0 from the graded situation to thelocal situation. The polynomiality consequence of the second result is ananalog of work by Kodiyalam on the behavior of Betti numbers of the minimalfree resolution of R/I^k over R.

Homology of powers of ideals: Artin--Rees numbers of syzygies and the Golod property