On the $h$-polynomials of cyclotomic standard graded commutative algebras

Abstract

We call a standard graded commutative $\mathbb{k}$-algebra cyclotomic if its $h$-polynomial has all its roots on the unit circle in the complex plane. Complete intersections provide typical examples of cyclotomic algebras, since the $h$-polynomial of any standard graded complete intersection is a product of polynomials of the form $1+t+\cdots +t^{m-1}$. We refer to such polynomials as being of type CI. A natural question is whether there exists a cyclotomic standard graded $\mathbb{k}$-algebra whose $h$-polynomial is not of type CI. In this paper, we give a partial answer to this question. We show that the $h$-polynomial $h_R(t)$ of a cyclotomic standard graded $\mathbb{k}$-algebra $R$ is of type CI whenever $h_R(1)\in \{1,4,6\}$ or $h_R(1)$ is prime. On the other hand, if $n\ge 8$ and $n$ is not prime, then there exists a cyclotomic standard graded $\mathbb{k}$-algebra $R$ whose $h$-polynomial $h_R(t)$ is not of type CI and satisfies $h_R(1)=n$.

14 pages