Related papers: Resolving sets and semi-resolving sets in finite p…

Motivated by a question of Erd\H{o}s on blocking sets in a projective plane that intersect every line only a few times, several authors have used unions of algebraic curves to construct such sets in $\mathbb{P}^2(\mathbb{F}_q)$. In this…

In this paper, we study the p-ary linear code C(PG(n, q)), q = p^h, p prime, h >= 1, generated by the incidence matrix of points and hyperplanes of a Desarguesian projective space PG(n, q), and its dual code. We link the codewords of small…

In this work, we prove the existence of maximal partial line spreads in PG(5,q) of size q^3+q^2+kq+1, with 1 \leq k \leq (q^3-q^2)/(q+1), k an integer. Moreover, by a computer search, we do this for larger values of k, for q \leq 7. Again…

Let $G$ be a transitive permutation group on a finite set with solvable point stabiliser and assume that the solvable radical of $G$ is trivial. In 2010, Vdovin conjectured that the base size of $G$ is at most 5. Burness proved this…

In this paper, we prove the degree upper bound of projective subschemes in terms of the reduction number and show that the maximal cases are only arithmetically Cohen-Macaulay subschemes with linear resolution. Furthermore, it can be shown…

In this paper, we give a finiteness criterion for the solutions of the sequence of semi-$q$-decomposable form equations and inequalities, where the semi-$q$-decomposable form is factorized into a family of $q$ nonconstant homogeneous…

In the previous works of the authors, a step-by-step algorithm FOP which uses any fixed order of points in the projective plane $\mathrm{PG}(2,q)$ is proposed to construct small complete arcs. In each step, the algorithm adds to a current…

A minimum storage regenerating (MSR) subspace family of $\mathbb{F}_q^{2m}$ is a set $\mathcal{S}$ of $m$-spaces in $\mathbb{F}_q^{2m}$ such that for any $m$-space $S$ in $\mathcal{S}$ there exists an element in $\mathrm{PGL}(2m, q)$ which…

We study the finite basis problem for $4$-element additively idempotent semirings whose additive reducts have two minimal elements and one coatom. Up to isomorphism, there are $112$ such algebras. We show that $106$ of them are finitely…

A resolving set for a graph $\Gamma$ is a collection of vertices $S$, chosen so that for each vertex $v$, the list of distances from $v$ to the members of $S$ uniquely specifies $v$. The metric dimension of $\Gamma$ is the smallest size of…

We show that the maximal number of planes in a complex smooth cubic fourfold in ${\mathbb P}^5$ is $405$, realized by the Fermat cubic only; the maximal number of real planes in a real smooth cubic fourfold is $357$, realized by the…

Let $R$ be a polynomial ring over a field and $I$ an ideal generated by three forms of degree three. Motivated by Stillman's question, Engheta proved that the projective dimension $\mathrm{pd}(R/I)$ of $R/I$ is at most 36, although the…

We prove that a non-empty set L of at most q^5+q^4+q^3+q^2+q+1 lines of PG(n, q) with the properties that (1) every point of PG(n,q) is incident with either 0 or q+1 elements of L, (2) every plane plane of PG(n, q) is incident with either…

We give explicit parametric equations for all irreducible plane projective sextic curves which have at most double points and whose total Milnor number is maximal (is equal to 19). In each case we find a parametrization over a number field…

We study two decomposition problems in combinatorial geometry. The first part deals with the decomposition of multiple coverings of the plane. We say that a planar set is cover-decomposable if there is a constant m such that any m-fold…

In this paper, we resolve a conjecture of Green and Liebeck [Disc. Math., 343 (8):117119, 2019] on codes in $PGL(2,q)$. To be specific, we show that: if $D$ is a dihedral subgroup of order $2(q+1)$ in $G=PGL(2,q)$, and $A=\{g\in G: g^{q+1}=…

A vertex set $U \subseteq V$ of an undirected graph $G=(V,E)$ is a \textit{resolving set} for $G$ if for every two distinct vertices $u,v \in V$ there is a vertex $w \in U$ such that the distance between $u$ and $w$ and the distance between…

Given a squarefree monomial ideal $I$ of a polynomial ring $Q$, we show that if the minimal free resolution $\mathbb{F}$ of $Q/I$ admits the structure of a differential graded (dg) algebra, then so does any ``pruning" of $\mathbb{F}$. In…

Let $G=(V,E)$ be a connected simple graph. The distance $d(u,v)$ between vertices $u$ and $v$ from $V$ is the number of edges in the shortest $u-v$ path. If $e=uv \in E$ is an edge in $G$ than distance $d(w,e)$ where $w$ is some vertex in…

In recent years, many useful applications of the polynomial method have emerged in finite geometry. Indeed, algebraic curves, especially those defined by R\'edei-type polynomials, are powerful in studying blocking sets. In this paper, we…