 

Hakop Hakopian and Sofik Toroyan
On the uniqueness of algebraic curves passing through nindependent nodes view print


Published: 
June 1, 2016 
Keywords: 
Algebraic curves, nindependent nodes, maximal curves, bivariate polynomial interpolation, GascaMaeztu conjecture. 
Subject: 
Primary: 41A05, 41A63; Secondary 14H50. 


Abstract
A set of nodes in the plane is called nindependent if for arbitrary data at
those nodes, there is a (not necessarily unique) polynomial of degree at most n that matches the given
information. We proved in a previous paper (HakopianToroyan, 2015) that the
minimal number of nindependent nodes determining uniquely the curve
of degree k≦ n passing through them equals to D:=(1/2)(k1)(2n+4k)+2. In this paper we bring a characterization of the case when at least
two curves of degree k pass through the nodes of an nindependent
node set of cardinality D1. Namely, we prove that the latter set has a very special construction: All its nodes but one belong to a (maximal) curve of degree k1.
We show that this result readily yields the above cited one. At the end, an important application to the GascaMaeztu conjecture is presented.


Author information
Hakop Hakopian:
Department of Informatics and Applied Mathematics, Yerevan State University, A. Manukyan Str. 1, 0025 Yerevan, Armenia
hakop@ysu.am
Sofik Toroyan:
Department of Informatics and Applied Mathematics, Yerevan State University, A. Manukyan Str. 1, 0025 Yerevan, Armenia
sofitoroyan@gmail.com

