Let $I$ be a nonzero proper monomial ideal in ${\rm k}[x_1,\ldots,x_r]$, and let $d$ be the maximal degree of a minimal monomial generator. We prove that the index of stability of the associated primes of its ordinary powers satisfies \[ {\rm astab}(I)\le\frac{(4e)^{r-1}}4\,d^{r-1}\qquad(r\ge3), \] while ${\rm astab}(I)=1$ for $r=2$. If $I$ has $s\ge2$ minimal generators, we also obtain \[ {\rm astab}(I)\le\min\{r,s\}(s-1)d^{\min\{r-1,s-2\}}. \] The first estimate answers Hoa's question on polynomial bounds. The proofs reduce stabilization to integer optimization for a matrix whose last row records the power of the ideal. We bound the degrees of its circuits and Graver basis elements and prove proximity estimates for optimal solutions in fixed fibers. A Steinitz rearrangement and a lattice-point count in the difference body of a simplex give the generator-independent bound. The matrix estimates hold for arbitrary indexed integral configurations in a simplex and real linear objective functions. The stability bounds hold over an arbitrary field, without normality or equigeneration assumptions.
Let $S=\mathbb{K}[x_1,\ldots,x_n]$ be a polynomial ring over a field $\mathbb{K}$. For a polymatroidal ideal $I\subseteq S$ generated in degree $d$, we prove $$\omega(I^{(s)})=\operatorname{reg} I^{(s)}=\operatorname{reg} I^s=sd\qquad(s\ge1),$$ where symbolic powers use all associated primes and $\omega$ denotes the la...
Let $I$ be a monomial ideal in a standard graded polynomial ring and let $\overline{I}$ denote its integral closure. We study the relationship between $\mathrm{v}(I)$ and $\mathrm{v}(\overline{I})$. We prove that $\mathrm{v}(\overline{I}) \leq \mathrm{v}(I)$ for monomial ideals in two variables, for equigenerated monom...
Let $R$ be a commutative ring with nonzero identity and let $R^\#$ denote the set of its nonzero nonunits. We extend the divisor topology $D(R)$, previously studied for integral domains, to arbitrary commutative rings and introduce the generalized divisor topology $GD(R)$ on $EC(R^\#)$. Its basic open sets are \[ B_a=\...
Suat Koç, İrem Doğan, Dilara Erdemir et al.· 0 citations
We study random polynomials of the form $R(x)=x^n+\omega_{n-1}x^{n-1}+\cdots+\omega_0$, where $\omega_0,\dots,\omega_{n-1}$ are independent, uniformly bounded integer-valued random variables, and $\omega_1,\dots,\omega_{n-1}$ have a fixed common law $\mu$. We prove (unconditionally) that, if the R\'{e}nyi entropy of or...
In this paper, we consider the asymptotic density of $T_c(x):=\#\{p\leq x:P^+(p-1)\geq p^c\}$, where $P^+(n)$ denote the largest prime factor of $n$. We show that for $x\rightarrow\infty$, for any $0<c\leq 1/2$, one has \begin{align*} \mathop{\lim\inf}_{x\rightarrow\infty}\frac{T_c(x)}{\pi(x)}&\geq\max\left(1-\frac{16}...
Let $R$ be a Noetherian standard $\mathbb{N}$-graded ring, and let $I_1,\ldots,I_r$ be homogeneous ideals of $R$. Let $L,M$, and $N$ be finitely generated $\mathbb{Z}$-graded $R$-modules with $N\subseteq M$. For $\underline{n} :=(n_1,\dots,n_r)\in\mathbb N^r$, set ${\bf I}^{\underline{n}}:=I_1^{n_1}\cdots I_r^{n_r}$. W...
