Geodesic
On the dimension of spaces of algebraic curves passing through $n$-independent nodes
Proceedings of the Yerevan State University. Physical and mathematical sciences, Tome 53 (2019) no. 2, pp. 91-100

Voir la notice de l'article provenant de la source Math-Net.Ru

Let the set of nodes $X$ in the plain be $n$-independent, i.e. each node has a fundamental polynomial of degree $n$. Suppose also that $|X|= (n+1)+n+\cdots+(n-k+4)+2$ and $3\leq k\leq n-1$. In this paper we prove that there can be at most 4 linearly independent curves of degree less than or equal to $k$ passing through all the nodes of $X$. We provide a characterization of the case when there are exactly four such curves. Namely, we prove that then the set $X$ has a very special construction: all its nodes but two belong to a (maximal) curve of degree $k-2$. At the end, an important application to the Gasca–Maeztu conjecture is provided.
Keywords: Algebraic curves, $n$-independent nodes, maximal curves
Mots-clés : Gasca–Maeztu conjecture. 
@article{UZERU_2019_53_2_a2,
     author = {H. A. Hakopian and H. M. Kloyan},
     title = {On the dimension of spaces of algebraic curves passing through $n$-independent nodes},
     journal = {Proceedings of the Yerevan State University. Physical and mathematical sciences},
     pages = {91--100},
     publisher = {mathdoc},
     volume = {53},
     number = {2},
     year = {2019},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/UZERU_2019_53_2_a2/}
}
TY  - JOUR
AU  - H. A. Hakopian
AU  - H. M. Kloyan
TI  - On the dimension of spaces of algebraic curves passing through $n$-independent nodes
JO  - Proceedings of the Yerevan State University. Physical and mathematical sciences
PY  - 2019
SP  - 91
EP  - 100
VL  - 53
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/UZERU_2019_53_2_a2/
LA  - en
ID  - UZERU_2019_53_2_a2
ER  - 
%0 Journal Article
%A H. A. Hakopian
%A H. M. Kloyan
%T On the dimension of spaces of algebraic curves passing through $n$-independent nodes
%J Proceedings of the Yerevan State University. Physical and mathematical sciences
%D 2019
%P 91-100
%V 53
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/UZERU_2019_53_2_a2/
%G en
%F UZERU_2019_53_2_a2
H. A. Hakopian; H. M. Kloyan. On the dimension of spaces of algebraic curves passing through $n$-independent nodes. Proceedings of the Yerevan State University. Physical and mathematical sciences, Tome 53 (2019) no. 2, pp. 91-100. http://geodesic.mathdoc.fr/item/UZERU_2019_53_2_a2/