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A stable property of Borel type ideals

Author: Cimpoeas Mircea
Publication venue
Publication date: 02/02/2007
Field of study

In this paper, we extend a result of Eisenbud-Reeves-Totaro in the frame of ideals of Borel type.Comment: 4 pages. to appear in "Comunications in Algebra

arXiv.org e-Print Archive

A note on the number of partitions of $n$ into $k$ parts

Author: Cimpoeas Mircea
Publication venue
Publication date: 01/11/2021
Field of study

We prove new formulas and congruences for

p(n,k):=

the number of partitions of

n

into

k

parts and

q(n,k):=

the number of partitions of

n

into

k

distinct parts. Also, we give lower and upper bounds for the density of the set

\{n\in\mathbb N\;:\;p(n,k)\equiv i(\bmod\; m)\}

, where

m\geq 2

and

0\leq i\leq m-1

.Comment: 8 page

arXiv.org e-Print Archive

A note on the linear independence of a class of series of functions

Author: Cimpoeas Mircea
Publication venue
Publication date: 19/02/2019
Field of study

For

k\in\mathbb R

, we consider a

\mathbb C

-algebra

\mathcal A_k

of holomorphic functions in the half plane

Re\; z>k

with (at most) subexponential growth on the real line to

+\infty

. In the

\mathcal A_k

-algebra of sequences of functions

\{\alpha:\mathbb N\rightarrow \mathcal A_k\}

, we consider the

\mathcal A_k

-subalgebra

\mathcal H_k

consisting in those

\alpha

for which there exists a continuous map

M:\{Re\; z>k\}\rightarrow [0,+\infty)

such that

|\alpha(n)(z)|\leq M(z)n^k

for all

Re\; z>k,n\geq 1

, and

\lim_{x\rightarrow +\infty}e^{-ax}M(x)=0

, for all

a>0

. Given

L

a sequence of holomorphic functions on

Re\; z>k

which satisfies certain conditions, we prove that the map

\alpha\mapsto F_L(\alpha)

, where

F_L(\alpha):=\sum_{n=1}^{+\infty}\alpha(n)(z)L(n)(z)

, is an injective morphism of

\mathcal A_k

-modules (or

\mathcal A_k

-algebras). Consequently, if

n\mapsto \alpha_j(n)(z)\in\mathbb C

1\leq j\leq r

, are linearly (algebraically) independent over

\mathbb C

, for

z

in a nondiscrete subset of

Re\; z>k

, then

F_{\alpha_1},\ldots,F_{\alpha_r}

are linearly (algebraically) independent over the quotient field of

\mathcal A_k

.Comment: 15 pages; minor corrections; to appear in The Journal of Analysi

arXiv.org e-Print Archive

A note on the action of Hecke groups on subsets of quadratic fields

Author: Cimpoeas Mircea
Publication venue
Publication date: 12/07/2020
Field of study

We study the action of the groups

H(\lambda)

generated by the linear fractional transformations

x:z\mapsto -\frac{1}{z} \text{ and }w:z\mapsto z+\lambda

, where

\lambda

is a positive integer, on the subsets

\mathbb Q^*(\sqrt{n})=\{\frac{a+\sqrt n}{c}\;|\;a,b=\frac{a^2-n}{c},c\in\mathbb Z\}

, where

n

is a square-free integer. We prove that this action has a finite number of orbits if and only if

\lambda=1

\lambda=2

, and we give an upper bound for the number of orbits for

\lambda=2

.Comment: 6 page

arXiv.org e-Print Archive

Polarization and spreading of monomial ideals

Author: Cimpoeas Mircea
Publication venue
Publication date: 20/12/2018
Field of study

We characterize the monomial ideals

I\subset K[x_1,\ldots,x_n]

with the property that the polarization

I^p

and

I^{\sigma^n}:=

the ideal obtained from

I

by the

n

-th iterated squarefree operator

\sigma

are isomorphic via a permutation of variables. We give several methods to construct such ideals. We also compare the depth and sdepth of

I

and

I^{\sigma^n}

.Comment: 19 page

arXiv.org e-Print Archive

On the semigroup ring of holomorphic Artin L-functions

Author: Cimpoeas Mircea
Publication venue
Publication date: 06/12/2018
Field of study

Let

K/\mathbb Q

be a finite Galois extension and let

\chi_1,\ldots,\chi_r

be the irreducible characters of the Galois group

G:=Gal(K/\mathbb Q)

. Let

f_1:=L(s,\chi_1),\ldots,f_r:=L(s,\chi_r)

be their associated Artin L-functions. For

s_0\in \mathbb C\setminus\{1\}

, we denote

Hol(s_0)

the semigroup of Artin

L

-functions, holomorphic at

s_0

. Let

\mathbb F

be a field with

\mathbb C \subseteq \mathbb F \subseteq \mathcal M_{<1}:=

the field of meromorphic functions of order

<1

. We note that the semigroup ring

\mathbb F[Hol(s_0)]

is isomorphic to a toric ring

\mathbb F[H(s_0)]\subseteq \mathbb F[x_1,\ldots,x_r]

, where

H(s_0)

is an affine subsemigroup of

\mathbb N^r

minimally generated by at least

r

elements, and we describe

\mathbb F[H(s_0)]

when the toric ideal

I_{H(s_0)}=(0)

. Also, we describe

\mathbb F[H(s_0)]

and

I_{H(s_0)}

when

f_1,\ldots,f_r

have only simple zeros and simple poles at

s_0

.Comment: 12 pages, some improvements over the first version (Theorem 1.6

arXiv.org e-Print Archive

On a generalization of monomial groups

Author: Cimpoeas Mircea
Publication venue
Publication date: 17/06/2022
Field of study

We study a class of finite groups, called almost monomial groups, which generalize the class of monomial groups and it is connected with the theory of Artin L-functions. Our method of research is based on finding similarities with the theory of monomial groups, whenever it is possible.Comment: 12 pages, minor correction

arXiv.org e-Print Archive

Some remarks on Borel type ideals

Author: Cimpoeas Mircea
Publication venue
Publication date: 07/09/2007
Field of study

We give new equivalent characterizations for ideals of Borel type. Also, we prove that the regularity of a product of ideals of Borel type is bounded by the sum of the regularities of those ideals.Comment: 5 page

arXiv.org e-Print Archive

On a Zeta-Barnes type function associated to graded modules

Author: Cimpoeas Mircea
Publication venue
Publication date: 30/03/2018
Field of study

Let

K

be a field and let

S=\bigoplus_{n\geq 0} S_n

be a positively graded

K

-algebra. Given

M=\bigoplus_{n\geq 0} M_n

, a finitely generated graded

S

-module, and

w>0

, we introduce the function

\zeta_M(z,w):= \sum_{n=0}^{\infty}\frac{H(M,n)}{(n+w)^z}

, where

H(M,n):=\dim_K M_n

n\geq 0

, is the Hilbert function of

M

, and we study the relations between the algebraic properties of

M

and the analytic properties of

\zeta_M(z,w)

. In particular, in the standard graded case, we prove that the multiplicity of

M

e(M)=(m-1)!\lim_{w\searrow 0}Res_{z=m}\zeta_M(z,w)

.Comment: 12 page

arXiv.org e-Print Archive

Stanley depth of square free Veronese ideals

Author: Cimpoeas Mircea
Publication venue
Publication date: 07/07/2009
Field of study

We compute the Stanley depth for the quotient ring of a square free Veronese ideal and we give some bounds for the Stanley depth of a square free Veronese ideal. In particular, it follows that both satisfy the Stanley's conjecture.Comment: 4 page

arXiv.org e-Print Archive