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

TL;DR

This paper investigates the roots of $h$-polynomials of cyclotomic standard graded algebras, showing conditions under which these polynomials are of a specific product form, and providing examples where they are not.

Contribution

It characterizes when the $h$-polynomial of a cyclotomic algebra is of type CI based on its value at 1, and constructs examples outside this class for certain values.

Findings

$h$-polynomials of cyclotomic algebras have roots on the unit circle.

If $h_R(1)$ is 1, 4, 6, or prime, then $h_R(t)$ is of type CI.

For $n otin ext{prime}$ with $n geq 8$, there exist cyclotomic algebras with $h$-polynomials not of type CI.

Abstract

We call a standard graded commutative $\Bbbk$-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 $\Bbbk$-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 $\Bbbk$-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 $\Bbbk$-algebra $R$ whose $h$-polynomial $h_R(t)$ is not of type CI and satisfies $h_R(1) = n$.