On the $h$-polynomials of cyclotomic standard graded commutative algebras
Akihiro Higashitani, Kenta Ueyama
TL;DR
This work investigates when the $h$-polynomial of a cyclotomic standard graded algebra has roots on the unit circle and factors as a product of $\Psi_m(t)$ (type CI). It proves that if the multiplicity $h_R(1)$ is in $\{1,4,6\}$ or is prime, then $h_R(t)$ is of type CI, tying cyclotomicity to hypersurface or regular structures; conversely, for every non-prime $n\ge 8$, it constructs cyclotomic algebras with $h_R(1)=n$ whose $h$-polynomials are not of type CI. The proofs blend Stanley–Reisner theory, the $g$-theorem, and Macaulay’s theorem, using unions and joins of simplicial complexes to build explicit counterexamples and to translate combinatorial data into realizable Hilbert functions. The results delineate the limits of the type CI characterization and reveal rich behavior of $h$-polynomials under cyclotomic constraints, with explicit constructions for small and large $n$ guiding the general theory.
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 $h_R(1) = n$.