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, s_1), . . . , (k_r,  s_r ), we are able to determine under which conditions there exists a t-spread strongly stable ideal I of S with β_{k_i ,k_i+ s_i (I ) = a_i , i = 1, . . . , r , as extremal Betti numbers, and then to construct it.