arXiv subjects
Publications and source records attributed to Mircea Cimpoeas.
Let $r$ and $n$ be two positive integers and $S=K[x_1,\ldots,x_{n+r-1}]$, the ring of polynomials in $n+r-1$ variable, over a field $K$. We consider the squarefree monomial ideal $I_{n,r}:= x_1 \cdots x_{r-1} (x_{r},\ldots,x_{r+n-1}) \subset S$ and we prove several results regarding the Hilbert depth of $S/I_{n,r}$. Also, we consider the special case $n=r$.
Let $n\geq m$ be two positive integers, $S_{n,m}=K[x_1,\ldots,x_n,y_1,\ldots,y_m]$ and $I_{n,m}=(x_iy_j\;:\;1\leq i\leq n,1\leq j\leq m)\subset S_{n,m}$ the edge ideal of a complete bipartite graph. Denote $h(n,m)=\operatorname{hdepth}(S_{n,m}/I_{n,m})$. We prove that $h(n,m)\geq \left\lceil \frac{n}{2} \right\rceil$ and the equality holds if $m$ belong to a certain interval centered in $\left\lceil \frac{n} {2} \right\rceil$. Also, we find some tight bounds for $h(n,n)$ and we prove several inequalities between $h(n,m)$ and $h(n,m')$.
Let $\mathbf a:=(a_1,\ldots,a_r)$ be a sequence of positive integers, $d\geq 2$ and $j\geq 1$, some integers. We study the functions $p_{\mathbf a,d}(n):=$ the number of integer solutions $(x_1,\dots,x_r)$ of $\sum_{i=1}^r a_ix_i=n$, with $x_i\geq 0$ and $x_i \equiv 0,1(\bmod\;d)$, for all $1\leq i\leq r$, and $p_{\mathbf a,d}(n;j):=$ the number of $(x_1,\ldots,x_r)$ as above which satisfy also the condition $\sum_{i=1}^r \left(x_i-(d-2)\left\lfloor \frac{x_i}{d} \right\rfloor\right) =j$. We give formulas for $p_{\mathbf a,d}(n)$ and its polynomial part $P_{\mathbf a,d}(n)$, and also for $p_{\mathbf a,d}(n;j)$. As an application, we compute the dimensions of the stable cohomology groups for certain line bundles associated to flag varieties, defined over an algebraically closed field of positive characteristic.
Let $S=K[x_1,\ldots,x_n]$ be the ring of polynomials in $n$ variables over an arbitrary field $K$. Given a finitely generated multigraded module $M$, its Stanley length, denoted by $\operatorname{slength}(M)$, is the minimal length of a Stanley decomposition of $M$. Let $I\subset S$ be a monomial ideal, minimally generated by $m$ monomials. We give an upper bound for $\operatorname{slength}(I)$, in terms of its minimal monomial generators. Also, we give precise formulas for $\operatorname{slength}(I)$, if $n=2$ or $m=2$. Also, we show that if $I$ has linear quotients, then $\operatorname{slength}(I)=m$, and the converse holds in some special cases.
We prove new formulas for $p_d(n)$, the number of $d$-ary partitions of $n$, and, also, for its polynomial part. Given a partition $\lambda=(\lambda_1,\ldots,\lambda_{\ell})$, its associated $j$-th symmetric elementary partition, $pre_{j}(\lambda)$, is the partition whose parts are $\{\lambda_{i_1}\cdots\lambda_{i_j}\;:\;1\leq i_1 < \cdots < i_j\leq \ell\}$. We prove that if $\lambda$ and $\mu$ are two $d$-ary partitions of length $\ell$ such that $pre_j(\lambda)=pre_j(\mu)$ and $\lambda_{i_1}\cdots \lambda_{i_j} = \mu_{i_1}\cdots \mu_{i_j}$, for all $1\leq i_1 < \cdots < i_j\leq \ell$, then $\lambda=\mu$.
We present a new combinatorial approach to the computation of the (real) Fourier expansions of $\cos^n(t)$ and $\sin^n(t)$, where $n\geq 1$ is an integer. As an application, we compute the Fourier expansions of $f(t)=\frac{1}{a-\cos t}$ and $g(t)=\frac{1}{a-\sin t}$, where $a\in\mathbb R$ with $|a|>1$.
Let $S_n=K[x_1,\ldots,x_n,y]$ and $I_n=(x_1y,x_2y,\ldots,x_ny)\subset S_n$ be the edge ideal of star graph. We prove that $\operatorname{hdepth}(S_n/I_n)\geq \left\lceil \frac{n}{2} \right\rceil + \left\lfloor \sqrt{n} \right\rfloor - 2$. Also, we show that for any $\varepsilon>0$, there exists some integer $A=A(\varepsilon)\geq 0$ such that $\operatorname{hdepth}(S_n/I_n)\leq \left\lceil \frac{n}{2} \right\rceil + \left\lfloor \varepsilon n \right\rfloor + A - 2$. We deduce that $\lim\limits_{n\to\infty} \frac{1}{n}\operatorname{hdepth}(S_n/I_n) = \frac{1}{2}$.
We study the Stanley depth and the Hilbert depth of the edge ideals of path graphs, cycle graphs, generalized star graphs and double broom graphs.
In this note, we introduce arithmetic Heilbronn supercharacters that generalize the notions of arithmetic Heilbronn characters and Heilbronn supercharacters and discuss several properties of them.
We introduce a new class of finite groups, called weak almost monomial, which generalize two different notions of "almost monomial" groups, and we prove it is closed under taking factor groups and direct products. Let $K/\mathbb Q$ be a finite Galois extension with a weak almost monomial Galois group $G$ and $s_0\in \mathbb C\setminus \{1\}$. We prove that Artin conjecture's is true at $s_0$ if and only if the monoid of holomorphic Artin $L$-functions at $s_0$ is factorial. Also, we show that if $s_0$ is a simple zero for some Artin $L$-function associated to an irreducible character of $G$ and it is not a zero for any other $L$-function associated to an irreducible character, then Artin conjecture's is true at $s_0$.
Let $I$ be a squarefree monomial ideal of $S=K[x_1,\ldots,x_n]$. We prove that if $\operatorname{hdepth}(S/I)\leq 8$ or $n\leq 10$ then $\operatorname{hdepth}(I)\geq \operatorname{hdepth}(S/I)-1$.
Let $I_{n,m} = (x_1\cdots x_{m},x_2 \cdots x_{m+1},\ldots,x_{n+1}x_{n+2}\cdots x_{n+m})$ be the $m$-path ideal of a path of length $n + m-1$ over a polynomial ring $S = \mathrm{k}[x_1,\ldots,x_{n+m}]$. We compute all the graded Betti numbers of all powers of $I_{n,m}$.
Let $J_{n,m} = (x_1\cdots x_{m},x_2 \cdots x_{m+1},\ldots,x_{n}x_1\cdots x_{m-1})$ be the $m$-path ideal of a cycle of length $n \ge 5$ over a polynomial ring $S = k[x_1,\ldots,x_n]$. Let $t\geq 1$ be an integer. We show that $J_{n,m}^t$ has a linear free resolution and give a precise formula for all of its Betti numbers when $m = n-1, n-2$.
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 $I$ be a monomial ideal of $S=K[x_1,\ldots,x_n]$. We show that the following are equivalent: (i) $I$ is principal, (ii) $\operatorname{hdepth}(I)=n$, (iii) $\operatorname{hdepth}(S/I)=n-1$. Assuming that $I$ is squarefree, we prove that if $\operatorname{hdepth}(S/I)\leq 3$ or $n\leq 5$ then $\operatorname{hdepth}(I)\geq \operatorname{hdepth}(S/I)+1$. Also, we prove that if $\operatorname{hdepth}(S/I)\leq 5$ or $n\leq 7$ then then $\operatorname{hdepth}(I)\geq \operatorname{hdepth}(S/I)$.
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.
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 $\operatorname{hdepth}(h_3)\leq 67$.
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 $n$.