Stanley's nonunimodal Gorenstein \(h\)-vector is optimal (Q2832802)

In the paper under review the authors study a longstanding open problem in combinatorial commutative algebra to provide a classification of all possible artinian Gorenstein \(h\)-vectors. They succeed to classify all possible \(h\)-vectors of graded artinian Gorenstein algebras in socle degree 4 and codimension \(\leq 17\), and in socle degree 5 and codimension \(\leq 25\). They obtain as a consequence that the least number of variables allowing the existence of a nonunimodal Gorenstein h-vector is 13 for socle degree 4, and 17 for socle degree 5. In particular, the smallest nonunimodal Gorenstein \(h\)-vector is \((1, 13, 12, 13, 1)\), which was constructed by \textit{R. P. Stanley} in his seminal paper on level algebras [Adv. Math. 28, 57--83 (1978; Zbl 0384.13012)]. All of these results are characteristic free.