The infinitesimal 16th Hilbert problem in the quadratic case

Let H(x,y) be a real cubic polynomial with four distinct critical values (in a complex domain) and let X _H =H _y -H _x be the corresponding Hamiltonian vector field. We show that there is a neighborhood ? of X _H in the space of all quadratic plane vector fields, such that any X∈? has at most two limit cycles. Oblatum 23-III-2000 & 19-VI-2000?Published online: 11 October 2000     

Weighted Maximum Over Minimum Modulus of Polynomials, Applied to Ray Sequences of Padé Approximants

Let a≥ 0 , ɛ >0 . We use potential theory to obtain a sharp lower bound for the linear Lebesgue measure of the set Here P is an arbitrary polynomial of degree ≤ n . We then apply this to diagonal and ray Padé sequences for functions analytic (or meromorphic) in the unit ball. For example, we show that the diagonal \left{ [n/n]\right} _n=1 ^∞ sequence provides good approximation on almost one-eighth of the circles centre 0 , and the \left{ [2n/n]\right} _n=1 ^∞ sequence on almost one-quarter of such circles. July 18, 2000. Date revised: . Date accepted: April 19, 2001.     

Leading coefficients of Kazhdan–Lusztig polynomials and fully commutative elements

Let W be a Coxeter group of type . We show that the leading coefficient, μ(x,w), of the Kazhdan–Lusztig polynomial P _x,w is always equal to 0 or 1 if x is fully commutative (and w is arbitrary).     

The entire solutions of a polynomial difference equation

Recently S. Shimomura has shown that the polynomial difference equationw(z + 1) =P(w(z)), whereP is a given polynomial of degree at least two, always has entire non-constant solutions. The present investigation shows how to construct all entire solutions of the equation and discusses some properties of the solutions.     

Hermite—Fejér Interpolation for Rational Systems

This paper considers Hermite—Fejér and Grünwald interpolation based on the zeros of the Chebyshev polynomials for the real rational system P _n (a ₁ , . . . , a _n ) with the nonreal poles in {a}ⁿ _k=1 C\[-1,1] paired by complex conjugation. This extends some well-known results of Fejér and Grünwald for the classical polynomial case. July 11, 1996. Dates revised: January 6, 1997 and July 30, 1997.     

Extrema of a Real Polynomial

In this paper, we investigate critical point and extrema structure of a multivariate real polynomial. We classify critical surfaces of a real polynomial f into three classes: repeated, intersected and primal critical surfaces. These different critical surfaces are defined by some essential factors of f, where an essential factor of f means a polynomial factor of f–c ₀, for some constant c ₀. We show that the degree sum of repeated critical surfaces is at most d–1, where d is the degree of f. When a real polynomial f has only two variables, we give the minimum upper bound for the number of other isolated critical points even when there are nondegenerate critical curves, and the minimum upper bound of isolated local extrema even when there are saddle curves. We show that a normal polynomial has no odd degree essential factors, and all of its even degree essential factors are normal polynomials, up to a sign change. We show that if a normal quartic polynomial f has a normal quadratic essential factor, a global minimum of f can be either easily found, or located within the interior(s) of one or two ellipsoids. We also show that a normal quartic polynomial can have at most one local maximum.     

The Ptolemy and Zb?ganu constants of normed spaces

In every inner product space H the Ptolemy inequality holds: the product of the diagonals of a quadrilateral is less than or equal to the sum of the products of the opposite sides. In other words, ‖x−y‖‖z−w‖≤‖x−z‖‖y−w‖+‖z−y‖‖x−w‖ for any points w,x,y,z in H. It is known that for each normed space (X,‖⋅‖), there exists a constant C such that for any w,x,y,z∈X, we have ‖x−y‖‖z−w‖≤C(‖x−z‖‖y−w‖+‖z−y‖‖x−w‖). The smallest such C is called the Ptolemy constant of X and is denoted by C_P(X). We study the relationships between this constant and the geometry of the space X, and hence with metric fixed point theory. In particular, we relate the Ptolemy constant C_P to the Zb?ganu constant C_Z, and prove that if X is a Banach space with , then X has (uniform) normal structure and therefore the fixed point property for nonexpansive mappings. We derive general lower and upper bounds for both C_P and C_Z, and calculate the precise values of these two constants for several normed spaces. We also present a number of conjectures and open problems.     

Sur la suite de polynômes orthogonaux associée à la formeu=δ c +λ(x−c) −1 L

We show that, ifL is regular, semi-classical functional, thenu is also regular and semi-classical for every complex λ, except for a discrete set of numbers depending onL andc. We give the second order linear differential equation satisfied by each polynomial of the orthogonal sequence associated withu. The cases whereL is either a classical functional (Hermite, Laguerre, Bessel, Jacobi) or a functional associated with generalized Hermite polynomials are treated in detail.

     

Growth in Poisson algebras

A criterion for polynomial growth of varieties of Poisson algebras is stated in terms of Young diagrams for fields of characteristic zero. We construct a variety of Poisson algebras with almost polynomial growth. It is proved that for the case of a ground field of arbitrary characteristic other than two, there are no varieties of Poisson algebras whose growth would be intermediate between polynomial and exponential. Let V be a variety of Poisson algebras over an arbitrary field whose ideal of identities contains identities {{x ₁, y ₁}, {x ₂, y ₂}, . . . , {x _m , y _m }} = 0 and {x ₁, y ₁} · {x ₂, y ₂} · . . . · {x _m , y _m } = 0, for some m. It is shown that the exponent of V exists and is an integer. For the case of a ground field of characteristic zero, we give growth estimates for multilinear spaces of a special form in varieties of Poisson algebras. Also equivalent conditions are specified for such spaces to have polynomial growth.     

The increase of sums and products dependent on (y 1, …,y n ) by rearrangement of this set

LetF(u, v) be a symmetric real function defined forα<u, v<β and assume thatG(u, v, w)=F(u, v)+F(u, w)−F(v, w) is decreasing inv andw foru≦min (u, v). For any set (y)=(y ₁, …,y _n ),α<y _i <β, given except in arrangement Σ _i ^=1/n F(y _i ,y _i+1) wherey _n+1=y ₁) is maximal if (and under some additional assumptions only if) (y) is arranged in circular symmetrical order. Examples are given and an additional result is proved on the productΠ _i ^=1/n [(y2i−1y2i) ^m +α ₁(y _2i−1 y _2i ) ^m−1+ … +a _m ] wherea _k ≧0 and where the set (y)=(y ₁, ..,y _n ),y _i ≧0 is given except in arrangement. The problems considered here arose in connection with a theorem by A. Lehman [1] and a lemma of Duffin and Schaeffer [2]. This paper is part of the author’s Master of Science dissertation at the Technion-Israel Institute of Technology. The author wishes to thank Professor B. Schwarz and Professor E. Jabotinsky for their help in the preparation of this paper.     

A discrepancy theorem concerning polynomials of best approximation inL w p [−1, 1]

Letw be a suitable weight function,B _n,p denote the polynomial of best approximation to a functionf inL _w ^p [–1, 1],v _n be the measure that associates a mass of 1/(n+1) with each of then+1 zeros ofB _n+1,p–B _n,p and be the arcsine measure defined by . We estimate the rate at which the sequencev _n converges to in the weak-^* topology. In particular, our theorem applies to the zeros of monic polynomials of minimalL _w ^p norm.This author gratefully acknowledges partial support from NSA contract #A4235802 during 1992, AFSOR Grant 226113 during 1993 and The Alexander von Humboldt Foundation during both of these years.     

Convergence of Rational Interpolants with Preassigned Poles

Let Ω be a domain in the extended complex plane such that ∞∈Ω . Further, let K= C / Ω and, for each n , let Q _n be a monic polynomial of degree n with all its zeros in K . This paper is concerned with whether (Q _n ) can be chosen so that, if f is any holomorphic function on Ω and P _n is the polynomial part of the Laurent expansion of Q _n f at ∞ , then (P _n /Q _n ) converges to f locally uniformly on Ω . It is shown that such a sequence (Q _n ) can be chosen if and only if either K has zero logarithmic capacity or Ω is regular. January 21, 1999. Date accepted: August 17, 1999.     

Harnack type inequalities for conformal scalar curvature equation

We derive a Harnack type inequality for the conformal scalar curvature equation on B _3R . If the positive scalar curvature function K(x) is sub-harmonic in a neighborhood of each critical point and the maximum of u over B _R is comparable to its maximum over B _3R , then the Harnack type inequality can be obtained. Zhang is supported by NSF-DMS-0600275.     

Hilbert curves of polarized varieties

Let X be a normal Gorenstein complex projective variety. We introduce the Hilbert variety V_X associated to the Hilbert polynomial χ(x₁L₁+?+x_ρL_ρ), where L₁,…,L_ρ is a basis of , ρ being the Picard number of X, and x₁,…,x_ρ are complex variables. After studying general properties of V_X we specialize to the Hilbert curve of a polarized variety (X,L), namely the plane curve of degree dim(X) associated to χ(xK_X+yL). Special emphasis is given to the case of polarized threefolds.     

Schwarz—Pick Inequalities for Derivatives of Arbitrary Order

   Abstract. Let Ω and Π be two simply connected domains in the complex plane C which are not equal to the whole plane C and let λ _Ω and λ _Π denote the densities of the Poincare metric in Ω and Π , respectively. For f: Ω → Π analytic in Ω , inequalities of the type

Deformations of Functions Without Real Critical Points

Abstract

Given a real analytic function f(x, y) with one critical point P ₀, we study deformations f _t of fsuch that, for any t ≠ 0, the analytic function f _t has no critical points in a neighborhood of P ₀. We give explicitly a deformation without real critical points for any function which has only one real branch with characteristic exponents (4, 2q, r).     

On the Location of Pseudozeros of a Complex Interval Polynomial

Given a univariate complex interval polynomial F, we provide a rigorous method for deciding whether there exists a pseudozero of F in a prescribed closed complex domain D. Here a pseudozero of F is defined to be a zero of some polynomial in F. We use circular intervals and assume that the boundary C of D is a simple curve and that C is the union of a finite number of arcs, each of which is represented by a rational function. When D is not bounded, we assume further that all the polynomials in F are of the same degree. Examples of such domains are the outside of an open disk and a half-plane with boundary. Our decision method uses the representation of C and the property that a polynomial in F is of degree 1 with respect to each coefficient regarded as a variable.      

Pseudozero Set of Real Multivariate Polynomials

The pseudozero set of a system P of polynomials in n variables is the subset of C ⁿ consisting of the union of the zeros of all polynomial systems Q that are near to P in a suitable sense. This concept arises naturally in Scientific Computing where data often have a limited accuracy. When the polynomials of the system are polynomials with complex coefficients, the pseudozero set has already been studied. In this paper, we focus on the case where the polynomials of the system have real coefficients and such that all the polynomials in all the perturbed polynomial systems have real coefficients as well. We provide an explicit definition to compute this pseudozero set. At last, we analyze different methods to visualize this set.      

Majorization of polynomials on the plane

We prove that for every χ[−1, 1] and every real algebraic polynomial f of degree n such that |f(t): 1 on [−1, 1], the following inequality takes place on the complex plane |f(x+iy)||T_n(1+iy)|,−∞y∞ where T_n is the Tchebycheff polynomial. This implies easily Vladimir Markov inequality.