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Silviu Balanescu

Publications and source records attributed to Silviu Balanescu.

14 recordsLinked to original sources
arXiv2025

On the Hilbert depth of the quotient ring of the edge ideal of a star graph

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 $\lim\limits_{n\to\infty} \frac{1}{n}\operatorname{hdepth}(S_n/I_n) = \frac{1}{2}$.

math.AC↗
arXiv2024

Graded Betti numbers of powers of path ideals of paths

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 $I_{n,m}$.

math.AC↗
arXiv2024

Betti numbers of powers of path ideals of cycles

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 $m = n-1, n-2$.

math.AC↗
arXiv2024

Remarks on the Hilbert depth of squarefree monomial ideals

Let $K$ be a infinite field, $S=K[x_1,\ldots,x_n]$ and $0\subset I\subsetneq J\subset S$ two squarefree monomial ideals. In a previous paper we proved a new formula for the Hilbert depth of $J/I$. In this paper, we illustrate how one can use the Stanley-Reisner correspondence between (relative) simplicial complexes and (quotients of) squarefree monomial ideals, in order to reobtain some basic properties of the Hilbert depth. More precisely, we show that $\operatorname{depth}(J/I)\leq \operatorname{hdepth}(J/I)\leq \dim(J/I)$. Also, we show that $\operatorname{hdepth}(I)\geq \operatorname{hdepth}(S/I)+1$, if $S/I$ is Cohen-Macaulay.

math.AC↗
arXiv2024

On the Hilbert depth of the Hilbert function of a finitely generated graded module

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 $\overline M=M\otimes_S S[x_{n+1}]$.

math.AC↗
arXiv2024

On the arithmetic Hilbert depth

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 $\operatorname{hdepth}(h)\leq 2^{n+1}$.

math.NT↗
arXiv2024

On the Hilbert depth of quadratic and cubic functions

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 $\operatorname{hdepth}(h_3)\leq 67$.

math.NT↗
arXiv2024

Several combinatorial inequalities related to squarefree monomial ideals

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 $f(t)=(1+t+\cdots+t^{m-1})^n$.

math.AC↗
arXiv2024

Depth and Stanley depth of powers of the path ideal of a cycle graph

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 $t\geq 1$.

math.AC↗
arXiv2024

Depth and Stanley depth of powers of the path ideal of a cycle graph. II

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 $t\geq 1$.

math.AC↗
arXiv2023

On the Hilbert depth of certain monomial ideals and applications

We study the Stanley depth and the Hilbert depth for $I$ and $S/I$, where $I\subset S=K[x_1,\ldots,x_N]$ is the intersection of monomial prime ideals with disjoint sets of variables. As an application, we obtain bounds for the Stanley depth of $I_{n,m}^t$ and $J_{n,m}^t$, where $I_{n,m}$ is the $m$-path ideal of the path graph of length $n$ and $J_{n,m}$ is the the $m$-path ideal of the cycle graph of length $n$.

math.AC↗
arXiv2023

On the Hilbert depth of monomial ideals

Let $S=K[x_1,\ldots,x_n]$ be the ring of polynomials over a field $K$. Given two monomial ideals $0\subset I\subsetneq J \subset S$, we present a new method to compute the Hilbert depth of $J/I$. As an application, we show that if $u\in S$ is a monomial regular of $S/I$, then $\operatorname{hdepth}(S/I)\geq \operatorname{hdepth}(S/(I,u))\geq \operatorname{hdepth}(S/I)-1.$ Also, we reprove the formula of the Hilbert depth of a squarefree Veronese ideal.

math.AC↗
arXiv2023

Depth and Stanley depth of powers of the path ideal of a path graph

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 $t\geq 1$.

math.AC↗