Graded Betti numbers of powers of ideals

Using the concept of vector partition functions, we investigate the asymptotic behavior of graded Betti numbers of powers of homogeneous ideals in a polynomial ring over a field. Our main results state that if the polynomial ring is equipped with a positive $\text{ZZ}$-grading, then the Betti numbers of powers of ideals are encoded by finitely many polynomials. More precisely, in the case of $\text{ZZ}$-grading, ${\text{ZZ}}^2$ can be splitted into a finite number of regions such that each region corresponds to a polynomial that depending to the degree $(\mu ,t)$, $\dim_k \left(\tor_i^S(I^t, k)_{\mu} \right)$isequaltooneofthesepolynomialsin$(\mu, t)$. This refines, in a graded situation, the result of Kodiyalam on Betti numbers of powers of ideals. Our main statement treats the case of a power products of homogeneous ideals in a ${\text{ZZ}}^d$-graded algebra, for a positive grading.