ON $n$-NODE LINES IN $GC_n$ SETS

An $n$-poised node set $\mathcal X$ in the plane is called $GC_n$ set, if the fundamental polynomial of each node is a product of linear factors. A line is called $k$-node line, if it passes through exactly $k$-nodes of $\mathcal X.$ At most $n+1$ nodes can be collinear in $\Xset$ and an $(n+1)$-node line is called maximal line. The well-known conjecture of M. Gasca and J.I. Maeztu states that every $GC_n$ set has a maximal line. Until now the conjecture has been proved only for the cases $n \le 5.$ In this paper we prove some results concerning $n$-node lines, assuming that the Gasca--Maeztu conjecture is true.

ON THE DIMENSION OF SPACES OF ALGEBRAIC CURVES PASSING THROUGH $ n $-INDEPENDENT NODES

Let the set of nodes $ \LARGE{x} $ in the plain be $ n $-independent, i.e., each node has a fundamental polynomial of degree $ n $. Suppose also that $ \vert \LARGE{x} \normalsize \vert \mathclose{=} (n \mathclose{+} 1) \mathclose{+} n \mathclose{+} \cdots \mathclose{+} (n \mathclose{-} k \mathclose{+} 4) \mathclose{+} 2 $ and $ 3 \mathclose{\leq} k \mathclose{\leq} n \mathclose{-} 1 $. 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 $ \LARGE{x} $. We provide a characterization of the case when there are exactly 4 such curves. Namely, we prove that then the set $ \LARGE{x} $ has a very special construction: all its nodes but two belong to a (maximal) curve of degree $ k \mathclose{-} 2 $. At the end, an important application to the Gasca-Maeztu conjecture is provided.

An Alternative Approach to Algebro-Geometric Solutions of the AKNS Hierarchy

We develop an alternative systematic approach to the AKNS hierarchy based on elementary algebraic methods. In particular, we recursively construct Lax pairs for the entire AKNS hierarchy by introducing a fundamental polynomial formalism and establish the basic algebro-geometric setting including associated Burchnall–Chaundy curves, Baker–Akhiezer functions, trace formulas, Dubrovin-type equations for analogs of Dirichlet and Neumann divisors, and theta function representations for algebro-geometric solutions.

UNIVERSALITY IN ORBIT SPACES OF SYMMETRY GROUPS AND IN SPONTANEOUS SYMMETRY BREAKING

Universality properties of orbit spaces of compact linear groups are proved and related to universality properties of patterns of spontaneous symmetry breaking. In particular, it is argued that the orbit spaces of all the compact linear groups possessing a basis of fundamental polynomial invariants with the same degrees can be classified in a finite (and small for small dimensions and degrees) number of isomorphism classes.