arXiv subjects
Publications and source records attributed to Hakop Hakopian.
We present a new formula for the Hermite multivariate interpolation problem in the framework of the Chung--Yao approach. By using the respective univariate interpolation formula, we obtain a direct and explicit solution to the classical partial fraction decomposition problem for rational functions, including the real case.
We characterize the sets of solvability for Hermite multivariate interpolation problems when the sum of multiplicities is at most $2n + 2$, with $n$ the degree of the polynomial space. This result extends an earlier theorem (2000) by one of the authors concerning the case $2n+1$. The latter theorem, in turn, can be regarded as a natural extension of a classical Theorem of Severi (1921).
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 pass through at most $n+1$ nodes of an $n$-correct set. An $(n+1)$-node line is called maximal line (C. de Boor, 2007). We say that a node $A\in\mathcal{X}$ uses a line $\ell,$ if $\ell$ is a factor of the fundamental polynomial of the node $A.$ Let $\mathcal{X}$ be an $n$-correct set. One of the main problems we study in this paper is to determine the maximum possible number of used $2$-node lines that share a common node $B \in\mathcal{X}.$ We show that this number equals $n$. Moreover, if there are $n$ such $2$-node lines, then $\mathcal{X}$ contains exactly $n$ maximal lines not passing through the common node $B$. Furthermore, if $\mathcal{X}$ is $GC_n$ set, there exists an additional maximal line passing through $B$. Hence, in this case, $\mathcal{X}$ has $n+1$ maximal lines and is Carnicer~Gasca set of degree $n$. Note that Carnicer~Gasca sets of degree $n$ with a prescribed set of $n$ used $2$-node lines can be readily constructed.
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 linearly independent curves of degree less than or equal to $n-1$ that pass through all the nodes of $\mathcal X.$ We provide a characterization of the case when there are exactly three such curves. Namely, we prove that then the set $\mathcal X$ has a special construction: either all its nodes belong to a curve of degree $n-2,$ or all its nodes but three belong to a (maximal) curve of degree $n-3.$ This result complements a result established recently by H. Kloyan, D. Voskanyan, and H. H. Note that the proofs of the two results are completely different.
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 linear factors. In 1982, M.~Gasca and J.~I.~Maeztu conjectured that any such set necessarily contains $n+1$ collinear nodes. So far, this had only been confirmed for $n\leq 5.$ In this paper, we make a step for proving the case $n=6.$
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.
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 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 $\mathcal X.$ We provide a characterization of the case when there are exactly four such curves. Namely, we prove that then the set $\mathcal X$ 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.
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 if the line is a factor of the fundamental polynomial of the node. 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 any $GC_n$ set and an $(n+1)$-node line is called a maximal line. The Gasca-Maeztu conjecture (1982) states that every $GC_n$ set has a maximal line. Until now the conjecture has been proved only for the cases $n \le 5.$ Here, for a line $\ell$ we introduce and study the concept of $\ell$-lowering of the set $\mathcal X$ and define so called proper lines. We also provide refinements of several basic properties of $GC_n$ sets regarding the maximal lines, $n$-node lines, the used lines, as well as the subset of nodes that use a given line.
In this paper, we prove the Noether theorem with the multiplicities described by PD operators. Despite the known analogue versions in this case the provided conditions are necessary and sufficient. We also prove the Cayley-Bacharach theorem with PD multiplicities. As far as we know this is the first generalization of this theorem in the case of multiple intersections.
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.
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 $Ax+By+C$ divides the fundamental polynomial of the node. 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 $GC_n$ sets and an $(n+1)$-node line is called maximal line. The Gasca - Maeztu conjecture (1982) states that every $GC_n$ set has a maximal line. Until now the conjecture has been proved only for the cases $n \le 5.$ Here we adjust and prove a conjecture proposed in the paper - V. Bayramyan, H. H., Adv Comput Math, 43: 607-626, 2017. Namely, by assuming that the Gasca-Maeztu conjecture is true, we prove that for any $GC_n$ set $\mathcal X$ and any $k$-node line $\ell$ the following statement holds: Either the line $\ell$ is not used at all, or it is used by exactly $\binom{s}{2}$ nodes of $\mathcal X,$ where $s$ satisfies the condition $σ:=2k-n-1\le s\le k.$ If in addition $σ\ge 3$ and $μ(\mathcal X)>3$ then the first case here is excluded, i.e., the line $\ell$ is necessarily a used line. Here $μ(\mathcal X)$ denotes the number of maximal lines of $\mathcal X.$ At the end, we bring a characterization for the usage of $k$-node lines in $GC_n$ sets when $σ=2$ and $μ(\mathcal X)>3.$
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 $\mathcal X.$ 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. Untill now the conjecture has been proved only for the cases $n \le 5.$ We say that a node uses a line if the line is a factor in the node's fundamental polynomial. It is a simple and well-known fact that any maximal line $M$ is used by all $\binom{n+1}{2}$ nodes in $\mathcal X\setminus M.$ Here we consider the main result of the paper - V. Bayramyan, H. Hakopian, On a new property of n-poised and $GC_n$ sets, Adv Comput Math, 43, (2017) 607-626, stating that any $n$-node line of $GC_n$ set is used either by exactly $\binom{n}{2}$ nodes or by exactly $\binom{n-1}{2}$ nodes, provided that the Gasca-Maeztu conjecture is true. In this paper we show that this result is not correct in the case $n=3.$ Namely, we bring an example of a $GC_3$ set and a $3$-node line there which is not used at all. Fortunately, then we were able to establish that this is the only possible counterexample, i.e., the above mentioned result is true for all $n\ge 1, n\neq 3.$ We also characterize the exclusive case $n=3$ and present some new results on the maximal lines and the usage of $n$-node lines in $GC_n$ sets.
There are several examples of spaces of univariate functions for which we have a characterization of all sets of knots which are poised for the interpolation problem. For the standard spaces of univariate polynomials, or spline functions the mentioned results are well-known. In contrast with this there are no such results in the bivariate case. As an exception one may consider only the Pascal classic theorem, in the interpolation theory interpretation. In this paper we consider a space of bivariate piecewise linear functions, for which we can readily find out whether the given node set is poised or not. The main tool we use for this purpose is the reduction by a basic subproblem, introduced in this paper.
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 ${\ell}$ is called $k$-node line for a node set $\mathcal X$ if it passes through exactly $k$ nodes. An $(n+1)$-node line is called maximal line. In 1982 M. Gasca and J. I. Maeztu conjectured that every $GC_n$-set possesses necessarily a maximal line. Till now the conjecture is confirmed to be true for $n \le 5$. It is well-known that any maximal line $M$ of $\mathcal X$ is used by each node in $\mathcal X\setminus M,$ meaning that it is a factor of the fundamental polynomial of each node. In this paper we prove, in particular, that if the Gasca-Maeztu conjecture is true then any $n$-node line of $GC_n$-set $\mathcal X$ is used either by exactly $\binom{n}{2}$ nodes or by exactly $\binom{n-1}{2}$ nodes. We prove also similar statements concerning $n$-node or $(n-1)$-node lines in more general $n$-poised sets. This is a new phenomenon in $n$-poised and $GC_n$ sets. At the end we present a conjecture concerning any $k$-node line.