Zero sets of bivariate Hermite polynomials

Zero sets of bivariate Hermite polynomials

J. Math. Anal. Appl. 421 (2015) 830–841 Contents lists available at ScienceDirect Journal of Mathematical Analysis and Applications www.elsevier.com...

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J. Math. Anal. Appl. 421 (2015) 830–841

Contents lists available at ScienceDirect

Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa

Zero sets of bivariate Hermite polynomials ✩ Iván Area a , Dimitar K. Dimitrov b,∗ , Eduardo Godoy c a

Departamento de Matemática Aplicada II, E.E. Telecomunicación, Universidade de Vigo, 36310-Vigo, Spain b Departamento de Matemática Aplicada, IBILCE, Universidade Estadual Paulista, 15054-000 São José do Rio Preto, SP, Brazil c Departamento de Matemática Aplicada II, E.E. Industrial, Universidade de Vigo, 36310-Vigo, Spain

a r t i c l e

i n f o

Article history: Received 29 May 2014 Available online 25 July 2014 Submitted by M.J. Schlosser Keywords: Bivariate Hermite polynomials Zero sets of bivariate polynomials Bivariate Gaussian distribution Bivariate orthogonal polynomials Hermite polynomials Algebraic plane curves

a b s t r a c t We establish various properties for the zero sets of three families of bivariate Hermite polynomials. Special emphasis is given to those bivariate orthogonal polynomials introduced by Hermite by means of a Rodrigues type formula related to a general positive definite quadratic form. For this family we prove that the zero set of the polynomial of total degree n + m consists of exactly n + m disjoint branches and possesses n + m asymptotes. A natural extension of the notion of interlacing is introduced and it is proved that the zero sets of the family under discussion obey this property. The results show that the properties of the zero sets, considered as affine algebraic curves in R2 , are completely different for the three families analyzed. © 2014 Elsevier Inc. All rights reserved.

1. Introduction The properties of the zeros of univariate orthogonal polynomials have been studied thoroughly because of their fundamental role as eigenvalues of Jacobi operators and important applications, such as nodes of Gaussian quadrature formulas. If {pn (x)} is a sequence of orthogonal polynomials on the real line, with respect to a positive Borel measure dμ(x), it is well known that all zeros of pn (x) are real, belong to the convex hull of the support of dμ(x), and are distinct. Moreover, the zeros of two consecutive polynomials pn (x) and pn+1 (x) interlace [8,18]. Despite the growing number of publications on multivariate orthogonal polynomials [4–7,11,20], there are few results on the zero sets of these polynomial families (see [11,14,19,22] and the references therein) and the lack of general results is due to two main reasons. First of all, there is a rich variety of polynomials ✩ Research supported by the Brazilian foundations CNPq under Grant 307183/2013-0 and FAPESP under Grants 2009/13832-9 and 2013/23606-1, and by the Ministerio de Economía y Competitividad of Spain under grant MTM2012-38794-C02-01, cofinanced by the European Community fund FEDER. The first author thanks the Departamento de Matemática Aplicada, IBILCE, Universidade Estadual Paulista, where the research on the paper was performed during his visit supported by FAPESP (Grant 2013/23606-1). * Corresponding author. E-mail addresses: [email protected] (I. Area), [email protected] (D.K. Dimitrov), [email protected] (E. Godoy).

http://dx.doi.org/10.1016/j.jmaa.2014.07.042 0022-247X/© 2014 Elsevier Inc. All rights reserved.

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of many variables, even if they are orthogonal with respect to a fixed Borel measure on Rd, because of the many possible choices of arrangements of the multivariate polynomials. Moreover, since the zero set of a multivariate polynomial is an algebraic variety, sometimes its study requires deep and fine results from algebraic geometry [16]. The interest on zeros of multivariate polynomials has been focused mainly on describing those families having common zeros, because this property is related to the existence of Gaussian cubature formulas [10,11,13,20–22]. In this paper we are interested in the properties of the zero sets of three families of bivariate Hermite polynomials considered as affine algebraic curves. Let Hne (x)

n x2 /2

= (−1) e

  dn  −x2 /2  x −n/2 √ e =2 Hn dxn 2

(1)

be the probabilistic Hermite polynomials, with zeros hen,k , k = 1, . . . , n, and Hn (x) be the Hermite polynomials (see [15, p. 10]): 2

Hn (x) = (−1)n ex

dn  −x2  e , dxn

n ≥ 0.

(2)

Consider the general bivariate Hermite polynomials represented as a sum of products of Hne in the form (see [3, p. 370, Eq. (21)])    m n (n−k)/2 k (m−k)/2 Hn,m (x, y; Λ) = (−1) k! b c a k k k=0     ax + by bx + cy e e √ √ Hm−k , × Hn−k a c 

min(n,m)

k

(3)

with a, c > 0, ac − b2 > 0, whose orthogonality and basic properties are provided in Section 3. The affine transformation ax + by √ , a

bx + cy √ c

(4)

   m n (n−k)/2 k (m−k)/2 e e (−1)k k! a b c Hn−k (s)Hm−k (t). k k

(5)

s=

t=

yields 

min(n,m)

ˆ n,m (s, t; Λ) = H

k=0

It is clear that the properties of the zero sets of Hn,m (x, y; Λ) can be easily recovered from those of ˆ n,m (s, t; Λ) via the transformation inverse to (4). Therefore we state our result for the zero sets (affine H algebraic plane curves)   ˆ n,m (s, t; Λ) = 0 Zn,m = (s, t) ∈ R2  H

(6)

ˆ n,m (s, t; Λ), with b = 0, as follows: of the polynomials H Theorem 1.1. The affine algebraic plane curve Zn,m defined in (6): • consists of exactly in n + m disjoint branches; • possesses n vertical asymptotes s = hen,k , k = 1, . . . , n, and m horizontal asymptotes t = hem,j , j = 1, . . . m.

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• If b > 0 (b < 0) then for every point (s0 , t0 ) ∈ Zn,m , there exists a function t = t(s) such that t(s0 ) = t0 ˆ n,m (s, t(s); Λ) = 0. Moreover, t(s) is a strictly decreasing (increasing) function of s. and H • If sign(α) = sign(b), every straight line t = αs + β intersects Zn,m at exactly n + m points. Moreover, the zero sets of two consecutive general bivariate Hermite polynomials (5) do not intersect, that is, Zn,m ∩ Zn−1,m = ∅,

Zn,m ∩ Zn,m−1 = ∅.

All these properties are illustrated in Figs. 1–4 below. There exist other generalizations of univariate Hermite polynomials to the bivariate case. Properties of the zero sets of two of these generalizations are given at the end of the paper. It turns out that they are rather different from those described in Theorem 1.1. 2. Some basic notions from algebraic geometry Let us recall some basic definitions and notions from algebraic geometry [16], especially concerning affine algebraic curves. We add some information on projective algebraic plane curves, and the precise formulation of the Bézout theorem in Section 5. We shall consider affine algebraic curves on R2 , that is Cp = {(x, y) ∈ R2 | p(x, y) = 0} where p is a bivariate nonconstant polynomial. The degree of the polynomial p is also called the degree of the curve. Since the polynomial ring R[x, y] is a unique factorization domain, any polynomial p has a unique factorization p = pn1 1 · · · pns s up to constant multiples, as a product of irreducible factors pi where the irreducible pi are nonproportional, i.e. pi = αpj with α ∈ R if i = j. In these conditions, the algebraic curve Cp defined by p(x, y) = 0 is the union of curves Cpi given by pi (x, y) = 0. A curve is said to be irreducible if its equation is an irreducible polynomial [16]. The point (x0 , y0 ) of an algebraic curve p(x, y) = 0 is said to be a singular point [16, p. 13] if, except for p(x0 , y0 ) = 0, it obeys ∂ ∂ p(x0 , y0 ) = p(x0 , y0 ) = 0. ∂x ∂y A curve is called nonsingular if all its points are nonsingular. Asymptotes to algebraic curves are usually defined in terms of the projective space. In the case of vertical or horizontal asymptotes that definition can be given in terms of affine curves Cp as follows. Suppose that p(x, y) = p0 (x) + p1 (x)y + · · · + pn (x)y n is the Taylor expansion of p(x, y) considered as a polynomial of y. If pn (x) possess real zeros x1 , . . . , xj , then the vertical lines x = xk , k = 1, . . . , j, are the vertical asymptotes of Cp . The horizontal asymptotes are determined in a similar way. The rigorous statement of the classical Bézout theorem is also given for plane projective curves. Its consequence for affine algebraic curves is that the number of common points of two such curves does not exceed the product of their degrees. 3. Basic properties of bivariate Hermite polynomials Let us consider the strictly positive definite quadratic form  t

2

2

ϕ(x, y) = ϕ(ω) = ω Λω = ax + 2bxy + cy ,

Λ=

a b b c

 , ω = (x y)t ,

(7)

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so that a, c > 0 and det(Λ) = ac − b2 > 0. Hermite [3, p. 373] defined the polynomials (1) by means of the Rodrigues type formula Hn,m (x, y; Λ) = (−1)n+m eϕ(x,y)/2

∂ n+m  −ϕ(x,y)/2  e , ∂xn ∂y m

n, m ≥ 0,

(8)

and proved that they are orthogonal in R2 with respect to the weigh function e−ϕ(x,y)/2 +∞

+∞

2 e−ϕ(x,y)/2 Hn,m (x, y; Λ)Hp,q (x, y; Λ) dx dy = ΩM δM,P ,

(9)

−∞ −∞

where M = n + m, P = p + q, δi,j denotes the Kronecker’s delta, and 2 ΩM

  j 2πn!an−k bk−j  2s j+s j−s j cj−s (−k)j−s (j − k + n − s + 1)s , = √ b (−1) a s ac − b2 s=0

(10)

where (A)j denotes the Pochhammer’s symbol. Further generalizations of (8) were given [1,2]. ˆ n,m (s, t; Λ) obtained from (3) by the transformation (4) satisfy the partial differential The polynomials H equation (see [12, p. 288, Eq. (10)] for a similar one)

  ∂2 ∂ b ∂2 ∂ ∂2 ˆ n,m (s, t; Λ) = 0. + +t + (n + m) H + 2√ − s ∂s2 ∂s ∂t ac ∂s∂t ∂t2

(11)

Moreover, the differential relations ˆ n,m (s, t; Λ) √ ∂H ˆ n−1,m (s, t; Λ), = n aH ∂s

ˆ n,m (s, t; Λ) √ ∂H ˆ n,m−1 (s, t; Λ), = m cH ∂t

(12)

hold and they imply √ √ ˆ n−1,m (s, t; Λ) + v2 m cH ˆ n,m−1 (s, t; Λ), ˆ n,m (s, t; Λ) = v1 n aH Dv H

(13)

ˆ n,m (s, t; Λ) obey the recurrence relations for any nonzero vector v = (v1 , v2 ). The polynomials H √ ˆ n,m (s, t; Λ) − naH ˆ n−1,m (s, t; Λ) − mbH ˆ n,m−1 (s, t; Λ), ˆ n+1,m (s, t; Λ) = s aH H √ ˆ n,m+1 (s, t; Λ) = t cH ˆ n,m (s, t; Λ) − nbH ˆ n−1,m (s, t; Λ) − mcH ˆ n,m−1 (s, t; Λ), H

(14) (15)

for n, m ≥ 1. 4. Proof of the main result Before we establish the general properties stated in Theorem 1.1, we show the simplest cases when either ˆ n,m (s, t; Λ) with respect to n and m we fix m to be n or m equals 0 or 1. Because of the symmetry of H equal to either 0 or 1. 4.1. The case when m = 0 From (5), we obtain   s n/2 e n/2 ˆ Hn,0 (s, t; Λ) = a Hn (s) = a Hn √ . 2

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ˆ 5,1 (s, t; Λ) coincides with the one of (16). It consists in 6 branches. The particular values are a = 2, b = 1, Fig. 1. The graph of H and c = 2.

ˆ n,0 (s, t; Λ) is simply a pencil of n vertical parallel lines Therefore, the zero set of the polynomial H s = hen,k ,

1 ≤ k ≤ n,

where hen,k are the zeros of the probabilistic Hermite polynomial Hne (x) defined in (1). 4.2. The case when m = 1 Now, using (5) we have e ˆ n,1 (s, t; Λ) = an/2 c1/2 Hne (s)H1e (t) − na(n−1)/2 bHn−1 (s) H e (s). = an/2 c1/2 Hne (s)t − na(n−1)/2 bHn−1

ˆ n,1 (s, t; Λ) = 0 is equivalent to Hence, the equation H t=

e nbHn−1 (s) . 1/2 1/2 a c Hne (s)

(16)

Since the polynomials in the numerator and denominator have positive leading coefficients and their zeros hen,k interlace, the partial fraction decomposition of the right-hand side is n  k=1

k , s − hen,k

where k are positive real numbers. In fact, it follows from the Lagrange interpolation formula that k =

e (hen,k ) Hn−1 d e e ds Hn (hn,k )

.

ˆ n,1 (s, t; Λ) = 0 coincides with the graph of (16), and it consists of Therefore, the affine algebraic curve H n + 1 branches as shown in Fig. 1. 4.3. The general situation We shall separate the proof of Theorem 1.1 into various lemmas. First we obtain a bivariate extension of the well-known interlacing property of the zeros of univariate orthogonal polynomials.

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Fig. 2. The zero sets of consecutive general bivariate Hermite polynomials do not intersect. The zero sets of the polynomials ˆ 3,2 (s, t; Λ) (dashed) and H ˆ 3,3 (s, t; Λ), with a = 1, b = −1, and c = 2 do not cross each other. H

Lemma 4.1. The zero sets of consecutive general bivariate Hermite polynomials (5) do not intersect. More precisely, Zn,m ∩ Zn−1,m = ∅,

Zn,m ∩ Zn,m−1 = ∅.

Proof. For symmetry reasons we establish only the first of the above statements. Assume that there exists ˆ n,m (s, t0 ; Λ) and pn−1 (s) = H ˆ n−1,m (s, t0 ; Λ) (s0 , t0 ) ∈ Zn,m ∩ Zn−1,m . Let us fix t0 and consider pn (s) = H as polynomials in the variable s. If follows from (12) that pn (s0 ) = pn (s0 ) = 0.

(17)

On the other hand, the partial differential equation (11) is a second order ordinary differential equation in the variable s, satisfied by pn (s). Since the coefficient of the second derivative of pn (s) in (11) is not zero, then the relation (17) implies that pn (s0 ) = 0. A consecutive differentiation with respect to s of the above (n) mentioned differential equation yields p n (s0 ) = · · · = pn (s0 ) = 0, which is not possible. This result is illustrated in Fig. 2. 2 Lemma 4.2. The affine algebraic curve Zn,m consists of at most n + m disjoint branches. Proof. We shall prove that the affine algebraic curve Zn,m contains no singular points. Let (s0 , t0 ) ∈ Zn,m ˆ n,m (s0 , t0 ; Λ) = 0 and both partial derivatives are also zero at (s0 , t0 ), i.e. be a singular point, that is H ∂ ˆ ∂ ˆ ˆ ∂s Hn,m (s0 , t0 ; Λ) = 0 and ∂t Hn,m (s0 , t0 ; Λ) = 0. From (13) we have that Hn−1,m (s0 , t0 ; Λ) = 0 and ˆ Hn,m−1 (s0 , t0 ; Λ) = 0, which is not possible because of Lemma 4.1. 2 Lemma 4.3. For any positive integer numbers n and m the affine algebraic curve Zn,m possess exactly n vertical asymptotes and m horizontal asymptotes. Proof. As it was mentioned above, the asymptotes of Zn,m can be computed from (5): it has n vertical asymptotes as s = hen,k , where hen,k , k = 1, . . . , n, are the n real zeros of Hne (x) defined in (1), and m horizontal asymptotes as t = hem,k , e (x). This property is shown in a particular case in where hem,k , k = 1, . . . , m, are the m real zeros of Hm Fig. 3. 2

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ˆ 2,3 (s, t; Λ) = 0, for a = 2, b = 1, c = 2, has two vertical asymptotes as s = he and three Fig. 3. The affine algebraic curve H 2,k horizontal asymptotes as t = he3,k .

As a consequence we obtain the following bivariate extension of the well-known property that the zeros of univariate orthogonal polynomials are simple and distinct. Corollary 4.4. The affine algebraic curve Zn,m consists of exactly n + m simple disjoint branches. Lemma 4.5. If b > 0 then for every point (s0 , t0 ) ∈ Zn,m , there exists a function t = t(s) such that t(s0 ) = t0 ˆ n,m (s, t(s); Λ) = 0. Moreover, t(s) is a strictly decreasing function of s. and H Proof. The proof goes by induction with respect to n + m, and it is an almost immediate consequence of the implicit function theorem combined with (12). Indeed, for b > 0 the statement is obviously true when n + m = 1 or 2. Then, second equation of (12) guarantees the existence, continuity and smoothness of t(s). Moreover, by (12) in a neighborhood of s0 , t (s) = −

√ ˆ ˆ n,m (s, t; Λ)/∂s n aH ∂H n−1,m (s, t; Λ) =− √ . ˆ ˆ ∂ Hn,m (s, t; Λ)/∂t m cHn,m−1 (s, t; Λ)

A detailed analysis of the sign on the right-hand side, which is based on the interlacing properties, shows that its sign is always positive. 2 Before we state our next result, we discuss some possibilities to define interlacing between two simple affine algebraic curves, f (x, y) = 0 and g(x, y) = 0. Suppose that f (x, y) = 0 divides R2 into r disjoint domains Uj , j = 1, . . . , r. The domains Ui and Uj will be called neighboring ones if the intersection of their closures is a branch of f (x, y) = 0. Then, we may say that the zero sets of f and g interlace if each Ui contains exactly one branch of g. A different way of defining interlacing is to add the requirement that g maintains a constant sign along every branch of f (x, y) = 0. The third possibility is to define interlacing of affine algebraic curves along lines. In this case, if is a line, we say that the affine algebraic curves f and g intersect along it if • the intersection of with f is a set of n distinct points, ∩ f = λj , j = 1, . . . , n, ordered in a natural way along the line , • ∩ g = μs , s = 1, . . . , n − 1, where μs are distinct points and ordered in the same manner, and • λ1 < μ1 < λ2 < μ2 < · · · < μn−1 < λn . This third one will be the notion of interlacing we adopt in our next statement.

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Fig. 4. Left figure illustrates the last statement of Theorem 1.1. The line t = αs + β intersects Zn,m at exactly n + m points when sign(α) = sign(b). On the picture a = c = 2, b = 1 and t = 2s. The right one is an example of a curve for which the same statement does not hold if the signs of α(= 1) and b(= −1) are opposite.

Lemma 4.6. If b > 0, every straight line t = αs + β with α > 0 intersects Zn,m at exactly n + m points.  = Zn,m ∩ . We shall prove by induction Proof. Let be the straight line t = αs + β with α > 0 and let Zn,m   with respect to n + m, except for the statement of the Lemma, that Zn,m interlaces with both Zn−1,m and  Zn,m−1 along . ˆ 1,0 (s, t; Λ) and H ˆ 0,1 (s, t; Λ) are either a horizontal or If n + m = 1, it was observed that the zero sets of H a vertical straight line, so that every line of the form t = αs + β with α > 0 intersects each of them exactly     once. Moreover, it is easy to observe that Z0,1 interlaces along with both Z1,1 and Z0,2 . Similarly, Z1,0   interlaces with both Z1,1 and Z2,0 along .    Let us assume that Zn,m possesses exactly n + m points and interlaces with both Zn−1,m and Zn,m−1 ˆ n,m (s, αs +β; Λ) is an/2 cm/2 αm > 0. Let ζ¯k be the points bealong . Observe that the leading coefficient of H  longing to Zn,m , arranged in decreasing order with respect to s. It follows from the induction hypothesis that these are exactly n + m distinct points on , ζ¯1 > ζ¯2 > · · · > ζ¯n+m . By the other statement of the induction ˆ n−1,m (s, αs + β; Λ) and H ˆ n,m−1 (s, αs + β; Λ) hypothesis, these points interlace with the zero sets of both H and we know that the leading coefficient of the latter polynomials are positive. Therefore,





ˆ n,m−1 (ζ¯k ; Λ) = (−1)k−1 , ˆ n−1,m (ζ¯k ; Λ) = sign H sign H

k = 1, . . . , n + m.

(18)

From (14) and (15) we immediately obtain 



ˆ n,m+1 (ζ¯k ; Λ) = (−1)k , ˆ n+1,m (ζ¯k ; Λ) = sign H sign H

k = 1, . . . , n + m,

(19)

ˆ n+1,m (s, αs + β; Λ) and because a, b and c are positive numbers. Since the leading coefficients of H ˆ Hn,m+1 (s, αs + β; Λ) are positive then ˆ n+1,m (s, αs + β; Λ) = lim H ˆ n,m+1 (s, αs + β; Λ) = +∞, lim H

s→+∞

s→+∞

ˆ n+1,m (s, αs + β; Λ) = lim H ˆ n,m+1 (s, αs + β; Λ) = (−1)n+m+1 (+∞). lim H

s→−∞

s→−∞

  The latter limit relations, together with (18) and (19), implies that Zn+1,m and Zn,m+1 contain exactly  n + m + 1 points and they interlace with Zn,m along . 2

Remark 1. If the signs of α and b are opposite the last statement of Theorem 1.1 is not true. An example is illustrated in Fig. 4.

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5. Projective curves and zero sets of another two families of bivariate Hermite polynomials The projective plane P2 [16, Section 1.6] is determined by points (ξ, η, ζ) in R3 , with (ξ, η, ζ) = (0, 0, 0), where two triples (ξ, η, ζ) and (ξ1 , η1 , ζ1 ) determine the same point if there exists κ ∈ R, κ = 0, such that ξ = κξ1 , η = κη1 , ζ = κζ1 . Every triple (ξ, η, ζ) which defines a point P is called a set of homogeneous coordinates of P , written P = (ξ : η : ζ). The inclusion R2 ⊂ P2 is defined by (x, y) → (x : y : 1). Then, the point (ξ : η : ζ) ∈ P2 with ζ = 0 corresponds to the point (ξ/ζ, η/ζ) ∈ R2 . The points of the complementary set ζ = 0 are called points at infinity. This notion is related to the choice of the coordinate ζ. An algebraic curve in P2 , or a projective algebraic plane curve is defined in homogeneous coordinates by an equation F (ξ, η, ζ) = 0, where F is a homogeneous polynomial. Then, whether F (ξ, η, ζ) = 0 holds or not is independent of the choice of the homogeneous coordinates of a point; that is, it is preserved on passing from (ξ1 , η1 , ζ1 ) to (κξ, κη, κζ), with κ = 0. A homogeneous polynomial F is also called a form. An affine algebraic curve of degree n with equation f (x, y) = 0 defines a homogeneous polynomial F (ξ, η, ζ) = ζ n f (ξ/ζ, η/ζ), and hence a projective curve with equation F (ξ, η, ζ) = 0. Vice versa, if the equation of the projective curve is F (ξ, η, ζ) = 0, then the equation of the corresponding affine curve is f (x, y) = 0, where f (x, y) = F (x, y, 1). Now we are ready to formulate the Bézout theorem [16] in its rigorous form: Theorem 1. Let X and Y be projective curves, with X nonsingular and not contained in Y . Then, the sum of the multiplicities of intersection of X and Y at all points of X ∩ Y equals the product of the degrees of X and Y . The above relation between projective and affine curves implies that two algebraic curves Cf and Cg , of degree m and n intersect at at most mn points in R2 . As we have already mentioned, classical univariate Hermite polynomials have been generalized to various different families of bivariate orthogonal polynomials. We discuss briefly the differences between the zero sets Zn,m of (5) established in Theorem 1.1 and those of two other families of bivariate Hermite polynomials. Despite that the zero sets of these two additional families are rather simple to determine and visualize, the differences are interesting if we look at the zero sets as affine algebraic or projective curves. 5.1. Product of two separated variables univariate Hermite polynomials Let us consider the bivariate polynomials defined by the product of two separated variables univariate Hermite polynomials (see [17, Chapter II]), that is Hn,m (x, y) = Hn (x)Hm (y), where the Hermite polynomials are defined in (2). It is easy to observe that the polynomials Hn,m (x, y) obey the Rodrigues formula

 ∂ n+m

 Hn,m (x, y) = (−1)n+m exp x2 + y 2 exp −x2 − y 2 , n m ∂x ∂y

(20)

so that they are a particular case of (8) with a = c = 2 and b = 0. The zero set of Hn,m (x, y) obviously consists of the n vertical lines x = hn,k , k = 1, . . . , n, and the m vertical lines y = hm,k , k = 1, . . . , m, where hr, are the zeros of the Hermite polynomial Hr (x). The polynomials Hn,m (x, y) and Hp,q (x, y) have pm +qn common zeros, assuming that |m −p| = 1 and |n −q| = 1 while, according to Bezout’s theorem, the maximal number of common points of two algebraic affine curves, of degrees m + n and p + q, is (m + n)(p + q). These properties are shown in Fig. 5. 5.2. Polar Hermite polynomials ∗ The second family of polynomials Hn,m (ρ, θ) of total degree 2n + m that we consider is defined by

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Fig. 5. The left graph corresponds to H3,3 (x, y) = H3 (x)H3 (y) = 0 defined by (20) and it consists of 3 vertical lines x = h3,k , k = 1, 2, 3, and 3 horizontal lines y = h3,k , k = 1, 2, 3. The graph on the right shows H3,3 (x, y) = 0 again, together with H4,2 (x, y) = 0 (dashed).

Fig. 6. Three plots of zero sets of polynomials, defined in (21), are shown. The left one shows the zero set of the polynomial with n = 3 and m = 0, given in Cartesian coordinates by − 16 (x2 + y 2 )3 + 32 (x2 + y 2 )2 − 3(x2 + y 2 ) + 1. In the middle, the zero set of the polynomial with n = 0 and m = 3 which is x(x2 − 3y 2 ). At the right, both curves are shown together.

2 ∗ Hn,m,c (ρ, θ) = ρm Lm cos(mθ), n ρ

2 ∗ Hn,m,s (ρ, θ) = ρm Lm sin(mθ), n ρ

(21)

where Lα n (x) are the Laguerre polynomials. A family of polynomials related to the latter have been used in [9] for obtaining analytic solutions of the heat equation. It follows from the definition that the entire family of polynomials is orthogonal in R2 with respect to w(x, y) = exp(−x2 − y 2 ). The zero set of each polynomial in this family is easy to describe: ∗ • The zero set of Hn,0 (ρ, θ) consists of n concentric circles which are disjoint closed curves, so that the ∗ corresponding algebraic curve is reducible, obviously without asymptotes. The zero sets of Hn,0 (ρ) and ∗ Hn+1,0 (ρ) “interlace”. ∗ (ρ, θ) consists of m “equally-spaced” straight lines and the corresponding curve is • The zero set of H0,m reducible in m linear terms. Moreover, (0, 0) is a zero of multiplicity m of the polynomial, so it is a singular point of the curve. ∗ (ρ, θ) consists of n concentric circles and m straight lines and the • If m, n > 0, the zero set of Hn,m corresponding curve is decomposed into n quadratic and m linear terms. Again, (0, 0) is a singular point of multiplicity m and there are additional 2nm singular points, namely the common ones of the n circles and the m lines.

These affine algebraic curves are shown in Figs. 6 and 7.

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∗ ∗ Fig. 7. Interlacing properties of zero sets of polynomials (21). On the left, the zero sets of H3,0 (ρ, θ) and H2,0 (ρ, θ) (dashed) interlace. ∗ ∗ On the right, the zero sets of H0,2 (ρ, θ) and H0,3 (ρ, θ) (dashed) have a multiple common point at the origin.

One may easily count the number of common points of two affine algebraic curves defined by ∗ ∗ Hn,m (ρ, θ) = 0 and Hr,t (ρ, θ) = 0 and this number varies depending on n, m, r and t. The only case when the Bézout theorem holds with the maximum number of common points, counting their multiplicity, which is equal to the product of the degrees of the polynomials, is when n = r = 0. In such a situation ∗ ∗ both curves H0,m (ρ, θ) = 0 and H0,t (ρ, θ) = 0 consist of collections of m and t straight lines, respectively. ∗ ∗ Moreover, it is possible that two such polynomials, say H0,m,s (ρ, θ) and H0,t,s (ρ, θ) posses a common line and this happens when one of the parameters m or t is a multiple of the other one. References [1] R. Aktaş, A. Altın, A generating function and some recurrence relations for a family of polynomials, in: Proceedings of the 12th WSEAS International Conference on Applied Mathematics, 2007, pp. 118–121. [2] A. Altın, R. Aktaş, A class of polynomials in two variables, Math. Morav. 14 (1) (2010) 1–14. [3] P. Appell, J. Kampé de Fériet, Fonctions hypergéométriques et hypersphériques. Polynomes d’Hermite, Gauthier-Villars, Paris, 1926, VII + 434 pp. [4] I. Area, N.M. Atakishiyev, E. Godoy, J. Rodal, Linear partial q-difference equations on q-linear lattices and their bivariate q-orthogonal polynomial solutions, Appl. Math. Comput. 223 (2013) 520–536. [5] I. Area, E. Godoy, On limit relations between some families of bivariate hypergeometric orthogonal polynomials, J. Phys. A 46 (035202) (2013), 11 pp. [6] I. Area, E. Godoy, J. Rodal, On a class of bivariate second-order linear partial difference equations and their monic orthogonal polynomial solutions, J. Math. Anal. Appl. 389 (2012) 165–178. [7] I. Area, E. Godoy, A. Ronveaux, A. Zarzo, Bivariate second-order linear partial differential equations and orthogonal polynomial solutions, J. Math. Anal. Appl. 387 (2) (2012) 1188–1208. [8] T.S. Chihara, An Introduction to Orthogonal Polynomials, Math. Appl., vol. 13, Gordon and Breach Science Publishers, New York, 1978. [9] D. Colton, J. Wimp, Analytic solutions of the heat equation and some formulas for Laguerre and Hermite polynomials, Complex Var. Theory Appl. 3 (4) (1984) 397–412. [10] F. Dai, Y. Xu, Approximation Theory and Harmonic Analysis on Spheres and Balls, Springer Monogr. Math., Springer, New York, 2013. [11] C.F. Dunkl, Y. Xu, Orthogonal Polynomials of Several Variables, Encyclopedia Math. Appl., vol. 81, Cambridge University Press, Cambridge, 2001. [12] A. Erdélyi, W. Magnus, F. Oberhettinger, F.G. Tricomi, Higher Transcendental Functions, vol. II, Robert E. Krieger Publishing Co. Inc., Melbourne, FL, 1981. [13] Z. Luo, Z. Meng, A result on common zeros location of orthogonal polynomials in two variables, J. Math. Anal. Appl. 324 (2) (2006) 785–789; Z. Luo, Z. Meng, J. Math. Anal. Appl. 329 (2) (2007) 1485 (Corrigendum). [14] Z. Luo, Z. Meng, F. Liu, Multivariate Stieltjes type theorems and location of common zeros of multivariate orthogonal polynomials, J. Math. Anal. Appl. 336 (1) (2007) 127–139. [15] A.F. Nikiforov, S.K. Suslov, V.B. Uvarov, Classical Orthogonal Polynomials of a Discrete Variable, Springer Ser. Comput. Phys., Springer-Verlag, Berlin, 1991. [16] I.R. Shafarevich, Basic Algebraic Geometry, vols. 1 and 2, Springer, Heidelberg, 2013. [17] P.K. Suetin, Orthogonal Polynomials in Two Variables, Anal. Methods Spec. Funct., vol. 3, Gordon and Breach Science Publishers, Amsterdam, 1999. [18] G. Szegő, Orthogonal Polynomials, 4th ed., Amer. Math. Soc. Colloq. Publ., vol. XXIII, American Mathematical Society, Providence, RI, 1975.

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