On Gorenstein sequences of socle degrees 4 and 5

We determine a new infinite class of symmetric h-vectors that cannot be Gorenstein sequences, a result left open in the works (Boji and Zanello, 2010 [4], Migliore et al. 2008 [17]). This includes the example h=(1,11,10,11,1), which was previously unknown. Combined with known results, this gives a complete characterization of the Gorenstein sequences for codimension ≤11 and socle degree 4. These results allow us to prove that every Gorenstein sequence of socle degree 5 and codimension ≤15 is unimodal, which improve the interesting results of [4,17] for socle degree 5. Furthermore, our results turn out to be independent of the characteristic of an infinite field k, while the results in [4] hold only when char(k)≠2.