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The aim of this paper is to connect the zeros of polynomials in two variables with the eigenvalues of a self-adjoint operator. This is done by use of a functional-analytic method. The polynomials in two variables are assumed to satisfy a five-term recurrence relation, similar to the three-term recurrence relation that the classical orthogonal polynomials satisfy.

Orthogonal polynomials in two or more variables (also multivariate polynomials) constitute a very old subject and have been investigated by many authors using various approaches. The usefulness and applications of classical orthogonal polynomials (COP) in one variable are very well-known and thus on its own this is a very strong motivation for generalizing several results of COP to multivariate polynomials. Moreover, the potential application of multivariate orthogonal polynomials in approximation techniques and numerical methods is another strong motivation.

For example, in numerical integration, the Gauss quadrature formula

There are various extensions in the literature of the COP to polynomials of several variables or polynomials of complex variables. For example, in [

The techniques used for the study of multivariate polynomials, orthogonal or not, constitute also a wide variety. In many cases the polynomials studied are constructed in such a way or are eventually eigenfunctions of certain partial differential operators. Furthermore, the properties of these polynomials or these operators are investigated. This is done, for example, in [

The topic of multivariate polynomials is quite vast and unfortunately there exist very few books. One such book is [

As mentioned in [

An especially interesting subject is the zeros of multivariate polynomials. However, the zeros of, for example, a polynomial in two variables are either single points or curves in the plane. Thus, results for such zeros could not have much in common with the results of the zeros of COP. However, if common zeros are taken into consideration, then there are many similarities with the one-dimensional case. Such a kind of result can be found in [

In this paper, a family of polynomials in two variables (2D-polynomials) of degrees

Instead of (

If one makes the conventions

Another motivation for considering (

Suppose two families of COP

For the study of the zeros of the polynomials

Following this philosophy, the zeros of

The authors believe that many interesting questions for the polynomials

The rest of the paper is organized as follows: In Section

Consider a finite dimensional Hilbert space

It is well-known that the zeros of the polynomials

The zeros

In order to generalize this in two dimensions, consider the finite dimensional Hilbert space

In this space, analogously to

Analogously to the operator

Obviously,

The spectrum of

The eigenvalues of

In this section, the zeros

Consider the 2D-polynomials defined by (

If the pair

then

with corresponding eigenvector

that is,

Conversely, if

By repeating the proof of Theorem

Assume that (A1) and (A2) hold. Then

there exist common zeros of

the following inequality holds:

where

(i) It follows from the fact that

(a) First of all

Indeed it is

(b) Suppose

In order to compute all the coefficients

(i) For

(ii) For

(iii) For

By induction it can be proved that for all

Analogously, one may choose

In this section, the zeros

diagonal elements

elements above the diagonal

elements below the diagonal

From (

Consider the 2D-polynomials defined by (

If the pair

Conversely, if (

(a) Notice that, in the case where

By comparing Theorems

The authors declare that there are no competing interests.