Graded algebras with cyclotomic Hilbert series

@article{Borzi2020GradedAW,
  title={Graded algebras with cyclotomic Hilbert series},
  author={Alessio Borzi and Alessio D'Al{\`i}},
  journal={arXiv: Commutative Algebra},
  year={2020}
}
Let $R$ be a positively graded algebra over a field. We say that $R$ is Hilbert-cyclotomic if the numerator of its reduced Hilbert series has all of its roots on the unit circle. Such rings arise naturally in commutative algebra, numerical semigroup theory and Ehrhart theory. If $R$ is standard graded, we prove that, under the additional hypothesis that $R$ is Koszul or has an irreducible $h$-polynomial, Hilbert-cyclotomic algebras coincide with complete intersections. In the Koszul case, this… Expand
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References

SHOWING 1-10 OF 29 REFERENCES
Hilbert functions of graded algebras
Let R be a Noetherian commutative ring with identity, graded by the nonnegative integers N. Thus the additive group of R has a direct-sum decomposition R = R, + R, + ..., where RiRi C R,+j and 1 E R,Expand
Koszul Algebras and Regularity
This is a survey paper on commutative Koszul algebras and Castelnuovo-Mumford regularity. Koszul algebras, originally introduced by Priddy, are graded K-algebras R whose residue field K has a linearExpand
On the growth of deviations
The deviations of a graded algebra are a sequence of integers that determine the Poincare series of its residue field and arise as the number of generators of certain DG algebras. In a sense,Expand
δ-Koszul Algebras
ABSTRACT Let A = A 0 ⊕ A 1 ⊕ A 2 ⊕ ··· be a graded K -algebra such that A 0 is a finite product of copies of the field K, A is generated in degrees 0 and 1,and dim K A 1 < ∞. We study those gradedExpand
Cyclotomic numerical semigroup polynomials with few irreducible factors
A numerical semigroup S is cyclotomic if its semigroup polynomial PS is a product of cyclotomic polynomials. The number of irreducible factors of PS (with multiplicity) is the polynomial length l(S)Expand
On the Hilbert function of a graded Cohen-Macaulay domain☆
Abstract A condition is obtained on the Hilbert function of a graded Cohen-Macaulay domain R = R 0 ⊛ R t ⊛ ⋯ over a field R0 = K when R is integral over the subalgebra generated by R1. A result ofExpand
Numerical Semigroups, Cyclotomic Polynomials, and Bernoulli Numbers
  • P. Moree
  • Computer Science, Mathematics
  • Am. Math. Mon.
  • 2014
TLDR
The intent of this paper is to better unify the various results within the cyclotomic polynomial and numerical semigroup communities. Expand
Combinatorial Commutative Algebra
Monomial Ideals.- Squarefree monomial ideals.- Borel-fixed monomial ideals.- Three-dimensional staircases.- Cellular resolutions.- Alexander duality.- Generic monomial ideals.- Toric Algebra.-Expand
Cyclotomic Numerical Semigroups
TLDR
The notion of cyclotomic exponents and polynomially related numerical semigroups is introduced and some properties are derived and some applications of these new concepts are given. Expand
Coefficients and higher order derivatives of cyclotomic polynomials: old and new.
The $n^{th}$ cyclotomic polynomial $\Phi_n(x)$ is the minimal polynomial of an $n^{th}$ primitive root of unity. Its coefficients are the subject of intensive study and some formulas are known forExpand
...
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2
3
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