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

Let $K$ be a field and let $S=K[x_1,dots,x_n]$ be a standard polynomial ring over a field $K$. We characterize the extremal Betti numbers, values as well positions, of a $t$-spread strongly stable ideal of $S$. Our approach is constructive. Indeed, given some positive integers $a_1,dots,a_r$ and some pairs of positive integers $(k_1,ell_1),dots,(k_r,ell_r)$, we are able to determine under which conditions there exist a $t$-spread strongly stable ideal $I$ of $S$ with $beta_{k_i, k_iell_i}(I)=a_i$, $i=1, ldots, r$, as extremal Betti numbers, and then to construct it.