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The total Betti numbers of the toric ideal of a simple graph are, in general, highly sensitive to any small change of the graph. In this paper we look at some combinatorial operations that cause total Betti numbers to change in predictable…
We show the asymptotic relationship between the limit of the normalized length function of a multi-$p-$family of ideals and that of its shifted family under linear growth conditions in a local domain of characteristic $p$. Examples of…
Let $R$ be a commutative noetherian local ring, and let $M$ be a finitely generated $R$-module. Inspired by works of Vasconcelos and Briggs on characterization of complete intersection local rings through the homological properties of the…
In this paper, we study the asymptotic behaviour of the v-number of a Noetherian graded filtration $\mathcal{I}= \{I_{[k]}\}_{k\geq 0}$ of a Noetherian $\mathbb{N}$-graded domain $R$. Recently, it is shown that $\mathrm{v}(I_{[k]})$ is…
In this paper, we study zero divisors in Hurwitz series rings and Hurwitz polynomial rings over general noncommutative rings. We first construct Armendariz rings that are not Armendariz of the Hurwitz series type and find various properties…
It is proved that a system of parameters is always a Q-sequence for several classes of mixed characteristic rings: rings in which the characteristic of the residue field is a nilpotent element, a mixed characteristic analogue of…
In this paper we settle some polynomial identity which provides a family of explicit Waring decompositions of any monomial $X_0^{a_0}X_1^{a_1}\cdots X_n^{a_n}$ over a field $\Bbbk$. This gives an upper bound for the Waring rank of a given…
Let $G$ be an additive abelian group. A sequence $S = g_1 \cdot \ldots \cdot g_{\ell}$ of terms from $G$ is a plus-minus weighted zero-sum sequence if there are $\varepsilon_1, \ldots, \varepsilon_{\ell} \in \{-1, 1\}$ such that…
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…
We introduce a detection algorithm for SAGBI basis in polynomial rings, analogous to a Gr\"obner basis detection algorithm previously proposed by Gritzmann and Sturmfels. We also present two accompanying software packages named…
A graded Artinian algebra $A$ has the Weak Lefschetz Property if there exists a linear form $\ell$ such that the multiplication map by $\ell:[A]_i\to [A]_{i+1}$ has maximum rank in every degree. The linear forms satisfying this property…
Given a prime number $p$ and a positive integer $m$, we provide a family of diagonal hypersurfaces $\{ f_n \}_{n = 1}^{\infty}$ in $m$ variables, for which the denominator of $\text{ fpt } (f_{n})$ (in lowest terms) is always $p$ and whose…
Let $I\subset R=K[x_1, \ldots, x_n]$ be a square-free monomial ideal, $\mathfrak{q}$ be a prime monomial ideal in $R$, $h$ be a square-free monomial in $R$ with $\mathrm{supp}(h) \cap (\mathrm{supp}(\mathfrak{q}) \cup…
We study the symbolic $F$-splitness of families of binomial edge ideals. We also study the strong $F$-regularity of the symbolic blowup algebras of families of binomial edge ideals. We make use of Fedder-like criteria and combinatorial…
Let $\xx= x_1,\ldots,x_r$ denote a system of elements of a commutative ring $R$. For an $R$-module $M$ we investigate when $\xx$ is $M$-pro-regular resp. $M$-weakly pro-regular as generalizations of $M$-regular sequences. This is done in…
In this paper, over a field of positive characteristic we exhibit an infinite family of counter examples to the Zariski Cancellation Problem in higher dimensions ($\geqslant 3$) which are pairwise non-isomorphic and also non-isomorphic to…
A Comprehensive Grobner system for a parametric ideal I in K(A)[X] represents the collection of all Grobner bases of the ideals I' in K[X] obtained as the values of the parameters A vary in K. The recent algorithms for computing them…
The Huneke-Wiegand conjecture is a decades-long open question in commutative algebra. Garc\'ia-S\'anchez and Leamer showed that a special case of this conjecture concerning numerical semigroup rings $\Bbbk[\Gamma]$ can be answered in the…
The theory of Rees algebras of monomial ideals has been extensively studied, and as a consequence, many (sometimes partial) equivalences between algebraic properties of monomial ideals, and combinatorial properties of simplicial complexes…
In this work, we extend the concept of the Lipschitz saturation of an ideal to the context of modules in some different ways, and we prove they are generically equivalent.