arXiv subjects
Publications and source records attributed to Mircea Cimpoeas.
Let $I$ be a squarefree monomial ideal of $S=K[x_1,\ldots,x_n]$. We prove that if $\operatorname{hdepth}(S/I)\leq 6$ of $n\leq 9$ then $\operatorname{hdepth}(I)\geq \operatorname{hdepth}(S/I)$, giving a positive answer to a problem putted in arxiv:2403.17078
Let $K$ be a field, $A$ a standard graded $K$-algebra and $M$ a finitely generated graded $A$-module. Inspired by our previous works, we study the Hilbert depth of $h_M$, that is $$\operatorname{hdepth}(h_M)=\max\{d\;:\; \sum\limits_{j\leq k} (-1)^{k-j} \binom{d-j}{k-j} h_{M}(j) \geq 0 \text{ for all } k\leq d\}, $$ where $h_M(-)$ is the Hilbert function of $M$, and we prove basic results regard it. Using the theory of hypergeometric functions, we prove that $\operatorname{hdepth}(h_S)=n$, where $S=K[x_1,\ldots,x_n]$. We show that $\operatorname{hdepth}(h_{S/J})=n$, if $J=(f_1,\ldots,f_d)\subset S$ is a complete intersection monomial ideal with $deg(f_i)\geq 2$ for all $1\leq i\leq d$. Also, we show that $\operatorname{hdepth}(h_{\overline M})\geq \operatorname{hdepth}(h_M)$ for any finitely generated graded $S$-module $M$, where $\overline M=M\otimes_S S[x_{n+1}]$.
Let $h:\mathbb Z \to \mathbb Z_{\geq 0}$ be a nonzero function with $h(k)=0$ for $k\ll 0$. We define the Hilbert depth of $h$ by $\operatorname{hdepth}(h)=\max\{d\;:\; \sum_{j\leq k} (-1)^{k-j}\binom{d-j}{k-j}h(j)\geq 0\text{ for all }k\leq d\}$. We show that $\operatorname{hdepth}(h)$ is a natural generalization for the Hilbert depth of a subposet $\operatorname{P}\subset 2^{[n]}$ and we prove some basic properties of it. Given $h(j)=\begin{cases} aj^n+b,& j\geq 0 \\ 0, & j<0 \end{cases}$, with $a,b,n$ positive integers, we compute $\operatorname{hdepth}(h)$ for $n=1,2$ and we give upper bounds for $\operatorname{hdepth}(h)$ for $n\geq 3$. More generally, if $h(j)=\begin{cases} P(j),& j\geq 0 \\ 0,& j<0 \end{cases}$, where $P(j)$ is a polynomial of degree $n$, with non-negative integer coefficients, and $P(0)>0$, we show that $\operatorname{hdepth}(h)\leq 2^{n+1}$.
Given a numerical function $h:\mathbb Z_{\geq 0}\to\mathbb Z_{\geq 0}$ with $h(0)>0$, the Hilbert depth of $h$ is $\operatorname{hdepth}(h)=\max\{d\;:\;\sum\limits_{j=0}^k (-1)^{k-j}\binom{d-j}{k-j}h(j)\geq 0\text{ for all }k\leq d\}$; see arXiv:2309.10521 . In this note, we study the Hilbert depth of the functions $h_2(j)=aj^2+bj+e$, $j\geq 0$, and $h_3(j)=aj^3+bj^2+cj+e$, $j\geq 0$, where $a,b,c,e$ are some integers with $a,e>0$. We prove that if $b<0$ and $b^2\leq 4ae$ then $\operatorname{hdepth}(h_2)\leq 11$, and, if $b<0$and $b^2>4ae$ then $\operatorname{hdepth}(h_2)\leq 13$. Also, we show that if $b<0$ and $b^2\leq 3ac$ then $\operatorname{hdepth}(h_3)\leq 67$.
Let $K$ be a field and $S=K[x_1,\ldots,x_n]$, the ring of polynomials in $n$ variables, over $K$. Using the fact that the Hilbert depth is an upper bound for the Stanley depth of a quotient of squarefree monomial ideals $0\subset I\subsetneq J\subset S$, we prove several combinatorial inequalities which involve the coefficients of the polynomial $f(t)=(1+t+\cdots+t^{m-1})^n$.
Let $\mathbf m=(x_1,\ldots,x_n)$ be the maximal graded ideal of $S:=K[x_1,\ldots,x_n]$. We present a new method for computing the Hilbert depth of powers of $\mathbf m$.
We prove new formulas for $\operatorname{DD}_k(n)$, the number of plane partition diamonds of length $k$ of $n$, and, also, for its polynomial part.
Let $J_{n,m}:=(x_1x_2\cdots x_m,\; x_2x_3\cdots x_{m+1},\; \ldots,\; x_{n-m+1}\cdots x_n,\; x_{n-m+2}\cdots x_nx_1, \ldots, x_nx_1\cdots x_{m-1})$ be the $m$-path ideal of the cycle graph of length $n$, in the ring $S=K[x_1,\ldots,x_n]$. Let $d=\gcd(n,m)$. We prove that $\operatorname{depth}(S/J_{n,m}^t)\leq d-1$ for all $t\geq n-1$. We show that $\operatorname{sdepth}(S/J_{n,n-1}^t)=\operatorname{depth}(S/J_{n,n-1}^t)=\max\{n-t-1,0\}$ for all $t\geq 1$. Also, we give some bounds for $\operatorname{depth}(S/J_{n,m}^t)$ and $\operatorname{sdepth}(S/J_{n,m}^t)$, where $t\geq 1$.
Let $J_{n,m}:=(x_1x_2\cdots x_m,\; x_2x_3\cdots x_{m+1},\; \ldots,\; x_{n-m+1}\cdots x_n,\; x_{n-m+2}\cdots x_nx_1, \ldots, x_nx_1\cdots x_{m-1})$ be the $m$-path ideal of the cycle graph of length $n$, in the ring of polynomials $S=K[x_1,\ldots,x_n]$. As a continuation of arxiv:2303.15032v2, we prove several new results regarding $\operatorname{depth}(S/J_{n,m}^t)$ and $\operatorname{sdepth}(S/J_{n,m}^t)$, where $t\geq 1$.
Using elementary methods, we prove new formulas for $\operatorname{pp}(n)$, the number of plane partitions of $n$, $\operatorname{pp}_r(n)$, the number of plane partitions of $n$ with at most $r$ rows, $\operatorname{pp}^s(n)$, the number of strict plane partitions of $n$ and $\operatorname{pp}^{so}(n)$, the number of symmetric plane partitions of $n$. Also, we give new formulas for $P_r(n)$, the number of $r$-component multipartitions of $n$.
Let $\mathbf a=(a_1,\ldots,a_r)$ be a sequence of positive integers and $k\geq 2$ an integer. We study $p_{k,\mathbf a}(n)$, the restricted $k$-multipartition function associated to $\mathbf a$ and $k$. We prove new formulas for $p_{k,\mathbf a}(n)$, its waves $W_j(n,k,\mathbf a)$'s and its polynomial part $P_{k,\mathbf a}(n)$. Also, we give a lower bound for the density of the set $\{n\geq 0\;:\;p_{k,\mathbf a}(n)\not\equiv 0(\bmod\;m)\}$, where $m\geq 2$ is an integer.
Let $I_{n,m}:=(x_1x_2\cdots x_m,\; x_2x_3\cdots x_{m+1},\; \ldots,\; x_{n-m+1}\cdots x_n)$ be the $m$-path ideal of the path graph of length $n$, in the ring $S=K[x_1,\ldots,x_n]$. We prove that: $$\mathtt{depth}(S/I_{n,m}^t)=\begin{cases} n-t+2 - \left\lfloor \frac{n-t+2}{m+1} \right\rfloor - \left\lceil \frac{n-t+2}{m+1} \right\rceil, & t \leq n+1-m \\ m-1,& t > n+1-m \end{cases},\text{ for all }t\geq 1.$$ Also, we prove that $\mathtt{depth}(S/I_{n,m}) \geq \mathtt{sdepth}(S/I_{n,m}^t) \geq \mathtt{depth}(S/I_{n,m}^t)$ and $\mathtt{sdepth}(I_{n,m}^t)\geq \mathtt{depth}(I_{n,m}^t)$, for all $t\geq 1$.
Let $Covering$ be the category of the category of fuzzy coverings, and $Partition$, the category of fuzzy partitions. We geometrically construct an isomorphism of categories between $Partition$ and a full subcategory of $Covering$, which can be used to derive bijections between fuzzy partitions and fuzzy coverings with finitely many sets. Also, we establish an isomorphism between $Covering[n]$, the category of coverings with $n$ fuzzy sets, and a subcategory of $Partition$, whose objects are partitions with $n$ sets which satisfy certain conditions, which can be also used to deduce another bijection between fuzzy partitions and fuzzy coverings with finitely many sets.
Given a quasi-monomial, respectively an almost monomial, group $A$ and a cyclic group $C$ of prime order $p>0$, we show that the wreath product $W=A\wr C$ is quasi-monomial (respectively almost monomial), if certain technical conditions hold.
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.
Given a finite group $G$, we study the monomial algebra $R_G$, generated by the monomial characters of $G$. In particular, we note that the integral closure of $R_G$ is contained in the algebra generated by those characters $χ$ for which their associated Artin L-function $L(s,χ)$ is holomorphic at $s_0\in\mathbb C\setminus\{1\}$. Also, we discuss the supercharacter theoretic case.
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 $0\leq i\leq m-1$.
We extend the notions of quasi-monomial groups and almost monomial groups, in the framework of supercharacter theories, and we study their connection with Artin's conjecture regarding the holomorphy of Artin $L$-functions.