arXiv subjects

Davit Voskanyan

Publications and source records attributed to Davit Voskanyan.

3 recordsLinked to original sources

arXiv2021

On plane algebraic curves passing through $n$-independent nodes

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 there are at most seven linearly independent curves of degree less than or equal to $k$ that pass through all the nodes of $\mathcal X.$ We provide a characterization of the case when there are exactly seven such curves. Namely, we prove that then the set $\mathcal X$ has a very special construction: all its nodes but three belong to a (maximal) curve of degree $k-3.$ Let us mention that in a series of such results this is the third one. In the end, an important application to the bivariate polynomial interpolation is provided, which is essential also for the study of the Gasca-Maeztu conjecture.

math.AG↗

arXiv2020

On a scale of criteria on $n$-dependence

In this paper we prove that a planar set $\mathcal{X}$ of at most $mn-1$ points, where $m \le n$, is $κ$-dependent, if and only if there exists a number r, $1 \le r \le m-1$, and an essentially $κ$-dependent subset $\mathcal{Y} \subset \mathcal{X}$, $\#\mathcal{Y} \ge rs$, where $r + s - 3 = κ$, belonging to an algebraic curve of degree $r$, and not belonging to any curve of degree less than $r$. Moreover, if $\#\mathcal{Y} = rs$ then the set $\mathcal{Y}$ coincides with the set of intersection points of some two curves of degrees $r$ and $s$, respectively. Let us mention that the first three criteria of the scale, for $m=1,2,3,$ are well-known results.

math.AG↗

arXiv2019

On the intersection points of two plane algebraic curves

We prove that a set $\mathcal X\subset \mathbb{C}^2,\ \#{\mathcal X}=mn,\ m\le n, $ is the set of intersection points of some two plane algebraic curves of degrees $m$ and $n,$ respectively, if and only if the following conditions are satisfied: a) Any curve of degree $m+n-3$ containing all but one point of $\mathcal X$, contains all of $\mathcal X,$ b) No curve of degree less than $m$ contains all of $\mathcal X.$ Let us mention that the conditions a) and b) in the "only if" direction of this result follow from the Ceyley-Bacharach and Noether theorems, respectively.

math.AG↗