Segre embeddings, Hilbert series and Newcomb's problem

Marcel Morales (1)

1 IF - Institut Fourier

Monomial ideals and toric rings are closely related. By consider a Grobner basis we can always associated to any ideal $I$ in a polynomial ring a monomial ideal ${\rm in}_\prec I$, in some special situations the monomial ideal ${\rm in}_\prec I$ is square free. On the other hand given any monomial ideal $I$ of a polynomial ring $S$, we can define the toric $K[I]\subset S$. In this paper we will study toric rings defined by Segre embeddings, we will prove that their $h-$ vectors coincides with the so called Simon Newcomb number's in probabilities and combinatorics. We solve the original question of Simon Newcomb by given a formula for the Simon Newcomb's numbers involving only positive integer numbers.

Mots clés: Segre ring; Grobner basis; Simon Newcomb' number; Hilbert series; Betti numbers

Domaines: Algèbre commutative [math.AC]; Géométrie algébrique [math.AG]

Marcel Morales. Segre embeddings, Hilbert series and Newcomb's problem. 2013. ⟨hal-00839652v2⟩