Related papers: On the basic properties of $GC_n$ sets

A planar node set $\mathcal X,$ with $|\mathcal X|=\binom{n+2}{2}$ is called $GC_n$ set if each node possesses fundamental polynomial in form of a product of $n$ linear factors. We say that a node uses a line $Ax+By+C=0$ if $Ax+By+C$…

An $n$-poised node set $\mathcal X$ in the plane is called $GC_n$ set if the (bivariate) fundamental polynomial of each node is a product of n linear factors. A line is called $k$-node line if it passes through exactly $k$-nodes of…

In this paper we consider n-poised planar node sets, as well as more special ones, called $GC_n$-sets. For these sets all $n$-fundamental polynomials are products of n linear factors as it always takes place in the univariate case. A line…

An $n$-correct set $\mathcal{X}$ in the plane is a set of nodes admitting unique interpolation with bivariate polynomials of total degree at most $n$. A $k$-node line is a line passing through exactly $k$ nodes of $\mathcal{X}.$ A line can…

An $n$-correct node set $\mathcal{X}$ is called $GC_n$ set if the fundamental polynomial of each node is a product of $n$ linear factors. In 1982 Gasca and Maeztu conjectured that for every $GC_n$ set there is a line passing through $n+1$…

A two-dimensional $n$-correct set is a set of nodes admitting unique bivariate interpolation with polynomials of total degree at most ~$n$. We are interested in correct sets with the property that all fundamental polynomials are products of…

An $n$-independent set in two dimensions is a set of nodes admitting (not necessarily unique) bivariate interpolation with polynomials of total degree at most $n.$ For an arbitrary $n$-independent node set $\mathcal X$ we are interested…

Suppose $\mathcal{X}$ is an $n$-correct set of nodes in the plane, that is, it admits a unisolvent interpolation with bivariate polynomials of total degree less than or equal to $n.$ Then an algebraic curve $q$ of degree $k\le n$ can pass…

Let a set of nodes $\mathcal X$ in plain be $n$-independent, i.e., each node has a fundamental polynomial of degree $n.$ Suppose also that $|\mathcal X|= d(n,k-2)+2,$ where $d(n,k-2) = (n+1)+n+\cdots+(n-k+4)$ and $\ k\le n-1.$ In this paper…

Let a set of nodes $\mathcal X$ in the plane be $n$-independent, i.e., each node has a fundamental polynomial of degree $n.$ Assume that $\#\mathcal X=d(n,k-3)+3= (n+1)+n+\cdots+(n-k+5)+3$ and $4 \le k\le n-1.$ In this paper we prove that…

A subset of vertices of a graph is minimal if, within all subsets of the same size, its vertex boundary is minimal. We give a complete, geometric characterization of minimal sets for the planar integer lattice X. Our characterization…

A set of nodes is called $n$-independent if each its node has a fundamental polynomial of degree $n.$ We proved in a previous paper [H. Hakopian and S. Toroyan, On the minimal number of nodes determining uniquelly algebraic curves, accepted…

Let K be a field of characteristic 0. We consider linear equations a1*x1+...+an*xn=1 in unknowns x1,...,xn from G, where a1,...,an are non-zero elements of K, and where G is a subgroup of the multiplicative group of non-zero elements of K.…

Let $(G_n(x))_{n=0}^\infty$ be a $d$-th order linear recurrence sequence having polynomial characteristic roots, one of which has degree strictly greater than the others. Moreover, let $m\geq 2$ be a given integer. We ask for…

For every $k>3$, we give a construction of planar point sets with many collinear $k$-tuples and no collinear $(k+1)$-tuples. We show that there are $n_0=n_0(k)$ and $c=c(k)$ such that if $n\geq n_0$, then there exists a set of $n$ points in…

Let a set of nodes $\mathcal X$ in the plane be $n$-independent, i.e., each node has a fundamental polynomial of degree $n.$ Assume that\\ $\#\mathcal X=d(n,n-3)+3= (n+1)+n+\cdots+5+3.$ In this paper we prove that there are at most three…

Centrality indices are used to rank the nodes of a graph by importance: this is a common need in many concrete situations (social networks, citation networks, web graphs, for instance) and it was discussed many times in sociology,…

A finite set $P$ of points in the plane is $n$-universal with respect to a class $\mathcal{C}$ of planar graphs if every $n$-vertex graph in $\mathcal{C}$ admits a crossing-free straight-line drawing with vertices at points of $P$. For the…

The purpose of this short problem paper is to raise an extremal question on set systems which seems to be natural and appealing. Our question is: which set systems of a given size maximise the number of $(n+1)$-element chains in the power…

A variety $X$ is covered by lines if there exist a finite number of lines contained in $X$ passing through each general point. I prove two theorems. Theorem 1:Let $X^n\subset P^M$ be a variety covered by lines. Then there are at most $n!$…