2026/04/30 by Francesca Cioffi, Margherita Guida, Enrica Pirozzi
Mathematics · #math.AC
Let Od denote the number of finite O-sequences of multiplicity d, namely the Hilbert functions of standard graded Artinian quotients of polynomial rings over a field. Starting from an iterative formula for computing Od, we pursue two complementary directions. First, letting Ad be the number of the finite O-sequences of multiplicity d whose last non-zero element is strictly larger than 1, we prove that the sequence (Ad+2)d≥ 1 is sub-Fibonacci. This result gives an enhancement of the sub-Fibonacci behavior of (Od)d≥ 1. Then, we provide a new algorithm for computing Od, with more efficient performances than other available algorithms. We use the computed data and statistical methods to obtain an empirical calibration, in the interval 1≤ d ≤ 1100, of the Stanley-Zanello asymptotic upper bound for log(Od) that better fits the observed values of log(Od). An analogous study of the Stanley-Zanello asymptotic lower bound for log(Od) is also carried out. The same method can be applied in every interval where the data are known. Some consequent prediction estimates are proposed. We also show that the sequence (Od/Od-1)d≥ 2 is strongly Cesàro convergent to 1. As a byproduct, we show that, if the sequence (Od/Od-1)d≥ 2 converges, then its limit must be equal to 1, thereby giving a negative answer to a question posed by L. G. Roberts in 1992 under the assumption of convergence.