Related papers: Symbolic powers, set-theoretic complete intersecti…
The main focus of these notes is recent work on linear systems in which line arrangements play a role, including problems such as semi-effectivity, containment problems of symbolic powers of homogeneous ideals in their powers, bounded…
We prove some inequalities regarding the Castelnuovo--Mumford regularity of symbolic powers and integral closure of powers of monomial ideals.
Symbolic powers are a classical commutative algebra topic that relates to primary decomposition, consisting, in some circumstances, of the functions that vanish up to a certain order on a given variety. However, these are notoriously…
We study the regularity of small symbolic powers and integral closure of small powers of edge ideals. We also prove that the regularity of integral closure of powers of edge ideals of graphs with at most two odd cycles is the same as the…
In this note we address the relation between symbolic and ordinary powers of the ideal of a reduced set or points in projective space: the so-called containment problem. In particular, we obtain sharp lower bounds on the Waldschmidt…
The symmetric tensor power of graphs is introduced and its fundamental properties are explored. A wide range of intriguing phenomena occur when one considers symmetric tensor powers of familiar graphs. A host of open questions are…
In this paper we explicitly describe the symbolic powers of curves ${\mathcal C}(q,m)$ in ${\mathbb P}^3$ parametrized by $( x^{d+2m}, x^{d+m} y^m, x^{d} y^{2m}, y^{d+2m})$, where $q,m$ are positive integers, $d=2q+1$ and $\gcd(d,m)=1$. The…
We study initial sequences of various configurations of planar points. We answer several questions asked in our previous paper (Symbolic powers of planar point configurations), and we extend our considerations to the asymptotic setting of…
We deal with the rigidity conjecture of symbolic powers over regular rings. This was asked by Huneke. Along with our investigation, we confirm a conjecture [7, Conjecture 3.8].
Let $G_{n,r}$ denote the graph with $n$ vertices $\{x_1,\ldots,x_n\}$ in cyclic order and for each vertex $x_i$ consider the set $A_i=\{x_{i-r},\ldots,x_{i-1},x_{i+1},x_{i+2},\ldots, x_{i+r}\},$ where $x_{i-j}$ is the vertex $x_{n+i-j}$,…
In this article we establish bounds for the Castelnuovo-Mumford regularity of projective schemes in terms of the degrees of their defining equations. The main new ingredient in our proof is to show that generic residual intersections of…
Let $G$ be a graph and let $I = I(G)$ be its edge ideal. When $G$ is unicyclic, we give a decomposition of symbolic powers of $I$ in terms of its ordinary powers. This allows us to explicitly compute the Waldschmidt constant and the…
Let $D$ be a weighted oriented graph with the underlying graph $G$ when vertices with non-trivial weights are sinks and $I(D), I(G) $ be the edge ideals corresponding to $D$ and $G,$ respectively. We give explicit description of the…
The article presents mathematical generalization of results which originated as solutions of practical problems, in particular, the modeling of transitional processes in electrical circuits and problems of resource allocation. However, the…
We explicitly compute the least degree of generators of all symbolic powers of the defining ideal of Fermat-like configuration of lines in $\mathbb{P}^3_\mathbb{C}$, except for the second symbolic powers, where we provide bounds for them.…
At present a number of current or proposed experiments are directed towards a search for a `new physics' by detecting variations of fundamental physical constants or violations of certain basic symmetries. Various problems related to the…
We use generalised Zeckendorf representations of natural numbers to investigate mixing properties of symbolic dynamical systems. The systems we consider consist of bi-infinite sequences associated with so-called random substitutions. We…
Research on power values of power sums has gained much attention of late, partially due to the explosion of refinements in multiple advanced tools in (computational) Number Theory in recent years. In this survey, we present the key tools…
This paper aims to initialize a dynamical aspect of symbolic integration by studying stability problems in differential fields. We present some basic properties of stable elementary functions and D-finite power series that enable us to…
The "finite intersection property" for bifunctions is introduced and its relationship with generalized monotonicity properties is studied. Some results concerning existence of solution for (quasi-)equilibrium problems are established and…