The Betti polynomials of powers of an ideal

For an ideal I in a regular local ring or a graded ideal I in the polynomial ring we study the limiting behavior of as k goes to infinity. By Kodiyalam’s result it is known that β_i(S/I^k) is a polynomial for large k. We call these polynomials the Kodiyalam polynomials and encode the limiting behavior in their generating polynomial. It is shown that the limiting behavior depends only on the coefficients on the Kodiyalam polynomials in the highest possible degree. For these we exhibit lower bounds in special cases and conjecture that the bounds are valid in general. We also show that the Kodiyalam polynomials have weakly descending degrees and identify a situation where the polynomials all have the highest possible degree.