A numerical characterization of the extremal Betti numbers of t-spread strongly stable ideals

Let K be a field and let S = K[x(1),..., x(n)] be a standard polynomial ring over a field K. We characterize the extremal Betti numbers, values as well as positions, of a t-spread strongly stable ideal of S. Our approach is constructive. Indeed, given some positive integers a(1),..., a(r) and some pairs of positive integers ((k)1, l(1)), ..., (k(r), l(r)), we are able to determine under which conditions there exists a t-spread strongly stable ideal I of S with beta(ki,ki+li) (I) = a(i), i = 1,..., r, as extremal Betti numbers, and then to construct it.