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Mircea Cimpoeas

Publications and source records attributed to Mircea Cimpoeas.

At least 55 records · Page 3Linked to original sources
arXiv2020

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

We study the action of the groups $H(λ)$ generated by the linear fractional transformations $x:z\mapsto -\frac{1}{z} \text{ and }w:z\mapsto z+λ$, where $λ$ 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 $λ=1$ or $λ=2$, and we give an upper bound for the number of orbits for $λ=2$.

math.GR↗
arXiv2019

Artin L-functions to almost monomial Galois groups

If $K/\mathbb Q$ is a finite Galois extension with an almost monomial Galois group and if $s_0\in\mathbb C\setminus\{1\}$ is not a common zero for any two Artin L-functions associated to distinct complex irreducible characters of the Galois group then all Artin L-functions of $K/\mathbb Q$ are holomorphic at $s_0$. We present examples and basic properties of almost monomial groups.

math.NT↗
arXiv2019

Two semigroup rings associated to a finite set of germs of meromorphic functions

We fix $z_0\in\mathbb C$ and a field $\mathbb F$ with $\mathbb C\subset \mathbb F \subset \mathcal M_{z_0}:=$ the field of germs of meromorphic functions at $z_0$. We fix $f_1,\ldots,f_r\in \mathcal M_{z_0}$ and we consider the $\mathbb F$-algebras $S:=\mathbb F[f_1,\ldots,f_r]$ and $\overline S:=\mathbb F[f_1^{\pm 1},\ldots,f_r^{\pm 1}]$. We present the general properties of the semigroup rings \begin{align*} & S^{hol}:=\mathbb F[f^{\mathbf a}:=f_1^{a_1}\cdots f_r^{a_r}: (a_1,\ldots,a_r)\in\mathbb N^r \text{ and }f^{\mathbf a}\text{ is holomorphic at }z_0],\\ & \overline S^{hol}:=\mathbb F[f^{\mathbf a}:=f_1^{a_1}\cdots f_r^{a_r}: (a_1,\ldots,a_r)\in\mathbb Z^r \text{ and }f^{\mathbf a}\text{ is holomorphic at }z_0], \end{align*} and we tackle in detail the case in which $\mathbb F=\mathcal M_{<1}$ is the field of meromorphic functions of order $<1$ and $f_j$'s are meromorphic functions over $\mathbb C$ of finite order with a finite number of zeros and poles.

math.CV↗
arXiv2019

On the restricted partition function via determinants with Bernoulli polynomials

Let $r\geq 1$ be an integer, $\mathbf a=(a_1,\ldots,a_r)$ a vector of positive integers and let $D\geq 1$ be a common multiple of $a_1,\ldots,a_r$. We prove that, if a determinant $Δ_{r,D}$, which depends only on $r$ and $D$, with entries consisting in values of Bernoulli polynomials is nonzero, then the restricted partition function $p_{\mathbf a}(n): = $ the number of integer solutions $(x_1,\dots,x_r)$ to $\sum_{j=1}^r a_jx_j=n$ with $x_1\geq 0, \ldots, x_r\geq 0$ can be computed in terms of values of Bernoulli polynomials and Bernoulli Barnes numbers.

math.NT↗
arXiv2019

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

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 $\{α:\mathbb N\rightarrow \mathcal A_k\}$, we consider the $\mathcal A_k$-subalgebra $\mathcal H_k$ consisting in those $α$ for which there exists a continuous map $M:\{Re\; z>k\}\rightarrow [0,+\infty)$ such that $|α(n)(z)|\leq M(z)n^k$ 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 $α\mapsto F_L(α)$, where $F_L(α):=\sum_{n=1}^{+\infty}α(n)(z)L(n)(z)$, is an injective morphism of $\mathcal A_k$-modules (or $\mathcal A_k$-algebras). Consequently, if $n\mapsto α_j(n)(z)\in\mathbb C$, $1\leq j\leq r$, are linearly (algebraically) independent over $\mathbb C$, for $z$ in a nondiscrete subset of $Re\; z>k$, then $F_{α_1},\ldots,F_{α_r}$ are linearly (algebraically) independent over the quotient field of $\mathcal A_k$.

math.NT↗
arXiv2019

On the restricted partition function via determinants with Bernoulli polynomials. II

Let $r\geq 1$ be an integer, $\mathbf a=(a_1,\ldots,a_r)$ a vector of positive integers and let $D\geq 1$ be a common multiple of $a_1,\ldots,a_r$. In a continuation of a previous paper we prove that, if $D=1$ or $D$ is a prime number, the restricted partition function $p_{\mathbf a}(n): = $ the number of integer solutions $(x_1,\dots,x_r)$ to $\sum_{j=1}^r a_jx_j=n$ with $x_1\geq 0, \ldots, x_r\geq 0$ can be computed by solving a system of linear equations with coefficients which are values of Bernoulli polynomials and Bernoulli Barnes numbers.

math.NT↗
arXiv2019

Determinants with Bernoulli polynomials and the restricted partition function

Let $r\geq 1$ be an integer, $\mathbf a=(a_1,\ldots,a_r)$ a vector of positive integers and let $D\geq 1$ be a common multiple of $a_1,\ldots,a_r$. We study two natural determinants of order $rD$ with Bernoulli polynomials and we present connections with the restricted partition function $p_{\mathbf a}(n):=$ the number of integer solutions $(x_1,\dots,x_r)$ to $\sum_{j=1}^r a_jx_j=n$ with $x_1\geq 0, \ldots, x_r\geq 0$.

math.NT↗
arXiv2018

Polarization and spreading of monomial ideals

We characterize the monomial ideals $I\subset K[x_1,\ldots,x_n]$ with the property that the polarization $I^p$ and $I^{σ^n}:=$ the ideal obtained from $I$ by the $n$-th iterated squarefree operator $σ$ 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 $I^{σ^n}$.

math.AC↗
arXiv2018

On the semigroup ring of holomorphic Artin L-functions

Let $K/\mathbb Q$ be a finite Galois extension and let $χ_1,\ldots,χ_r$ be the irreducible characters of the Galois group $G:=Gal(K/\mathbb Q)$. Let $f_1:=L(s,χ_1),\ldots,f_r:=L(s,χ_r)$ 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 $s_0$.

math.NT↗
arXiv2018

Independence of Artin L-functions

Let $K/\mathbb Q$ be a finite Galois extension. Let $χ_1,\ldots,χ_r$ be $r\geq 1$ distinct characters of the Galois group with the associated Artin L-functions $L(s,χ_1),\ldots, L(s,χ_r)$. Let $m\geq 0$. We prove that the derivatives $L^{(k)}(s,χ_j)$, $1\leq j\leq r$, $0\leq k\leq m$, are linearly independent over the field of meromorphic functions of order $<1$. From this it follows that the L-functions corresponding to the irreducible characters are algebraically independent over the field of meromorphic functions of order $<1$.

math.NT↗
arXiv2018

On a Zeta-Barnes type function associated to graded modules

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 $ζ_M(z,w):= \sum_{n=0}^{\infty}\frac{H(M,n)}{(n+w)^z}$, 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 $ζ_M(z,w)$. In particular, in the standard graded case, we prove that the multiplicity of $M$, $e(M)=(m-1)!\lim_{w\searrow 0}Res_{z=m}ζ_M(z,w)$.

math.AC↗
arXiv2018

On the restricted partition function II

Let $\mathbf a=(a_1,\ldots,a_r)$ be a vector of positive integers. In continuation of a previous paper we present other formulas for the restricted partition function $p_{\mathbf a}(n): = $ the number of integer solutions $(x_1,\dots,x_r)$ to $\sum_{j=1}^r a_jx_j=n$ with $x_1\geq 0, \ldots, x_r\geq 0$.

math.CO↗
arXiv2018

On the restricted partition function

For a vector $\mathbf a=(a_1,\ldots,a_r)$ of positive integers we prove formulas for the restricted partition function $p_{\mathbf a}(n): = $ the number of integer solutions $(x_1,\dots,x_r)$ to $\sum_{j=1}^r a_jx_j=n$ with $x_1\geq 0, \ldots, x_r\geq 0$ and its polynomial part.

math.CO↗
arXiv2016

On the Stanley depth of powers of some classes of monomial ideals

Given arbitrary monomial ideals $I$ and $J$ in polynomial rings $A$ and $B$ over a field $K$, we investigate the Stanley depth of powers of the sum $I+J$, and their quotient rings, in $A\otimes_K B$ in terms of those of $I$ and $J$. Our results can be used to study the asymptotic behavior of the Stanley depth of powers of a monomial ideal. For instance, we solved the case of monomial complete intersection.

math.AC↗