Polynomials are the quintessential building blocks of mathematics. The Fundamental Theorem of Algebra tells us that a polynomial p(z)=a₀ + a₁z + … + a_nzⁿ of degree n has exactly n roots (zeros), counting multiplicities. If the degree n ≥ 5, then, in general, there is no formula--such as the quadratic formula--to solve a quadratic equation for the explicit determination of the roots of the polynomial p(z). In contemporary parlance, polynomials that have only real roots (zeros) are called hyperbolic polynomials. The appelation "hyperbolic" was introduced by Lars Gårding (c. 1958) in the context of certain equations called hyperbolic partial differential equations. These ideas are connected with matrix theory: recall that the characteristic polynomial associated with a real, symmetric square matrix A is a hyperbolic polynomial and whence the eigenvalues of A are all real.

In calculus, we frequently encounter questions about the location and nature of the zeros of polynomials having real coefficients. If p(x) is hyperbolic, then, it follows from Rolle's theorem that p'(x) is also hyperbolic. In other words, when the linear operator, D = d/dx, is applied to a polynomial p(x), all of whose roots lie on the x-axis, then all the roots of the polynomial p'(x) also lie on the x-axis. If the coefficients of p(z) are complex numbers, then a classical result (known as the Gauss-Lucas Theorem) assures us that the zeros of p'(z) lie in the region which is common to all the half-planes containing the roots of p(z). Now, in dealing with polynomials, we can think of the differentiation operator D = d/dx as giving rise to the sequence T = {0,1,2, … } which preserves hyperbolicity. Are there other sequences or more general linear operators which preserve hyperbolicity?

In 1914, inspired, in part, by the work of Edmond Laguerre (1834-1886), in a celebrated paper, entitled Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen, George Póya and Issai Schur completely characterized hyperbolicity preserving sequences. These authors called such sequences multiplier sequences (of the first kind). To clarify this notion, set T={γ₀,γ₁, γ₂, … } and define T[x^k]=γ_k x^k, k=0,1,2…, and T[p(x)]=T[∑₀ⁿ a_kx^k] = ∑₀ⁿ γ_k a_k x^k = q(x). Thus T is a multiplier sequence if and only if q(x) is hyperbolic, whenever p(x) is hyperbolic. By way of illustration, the reader may wish to show that if q(x) is a hyperbolic polynomial with only negative roots, then the sequence T={ q(k) }_k=0^∞ is a multiplier sequence.

Pólya and Schur provided both algebraic and transcendental characterizations of multiplier sequences. The latter characterization states that all multiplier sequences are generated by the Taylor coefficients of (entire) functions which are (uniform) limits of hyperbolic polynomials. The algebraic description of multiplier sequences asserts that we need only to "apply" the sequence T to a particular sequence of (test) polynomials. More precisely, T={γ₀, γ₁, γ₂, …} is a multiplier sequence if and only if the polynomial T[(1+x)ⁿ] is hyperbolic for every positive integer n. R. P. Boas hailed this main theorem as "a key result on the boundary between Algebra and Analysis."

A Sample Open Problem and the Ubiquity of Hyperbolicity

Mathematicians have the propensity to formulate problems which are sufficiently general so as to encompass many special cases. It is in this spirit, and in light of the numerous special results in the literature dealing with various linear operators and the distribution of zeros of polynomials, that in 2004 the following open problem was proposed. In order to state the problem, let π_n denote the vector space (over R or C) of all polynomials of degree ≤ n. For Ω ⊆ C (where Ω is an appropriate set of interest), let π_n(Ω) denote the class of all polynomials of degree ≤ n all of whose zeros lie in Ω and let π(Ω) be the class of all polynomials all of whose zeros are in Ω.

Problem: Describe all the linear operators T which map (transform) polynomials of degree ≤ n all of whose zeros lie in Ω into polynomials all of whose zeros lie Ω. That is, characterize all linear operators

T: π_n(Ω) → π(Ω)∪ {0} for n=2, 3, 4, … .

Recently, J. Borcea, P. Brändén and B. Shapiro solved the above problem in the case when Ω= R and obtained multivariate extensions for all finite-order linear differential operators with polynomial coefficients. Their method is based, in part, on the analysis of hyperbolic polynomials in several variables. The "test" polynomials (1+x)ⁿ mentioned above are now replaced by the complex polynomials (z+w)ⁿ in two variables. The above problem, however, is still unsolved in a number of important cases; as for example, when Ω is a strip in the complex plane.

There are some natural extensions of these problems to rational functions as well as transcendental entire functions represented by certain integrals. It is such generalizations that motivated (nay, provoked) a number group discussions dealing with questions related to what is arguably the most famous unsolved problem in mathematics: the Riemann Hypothesis. There are many equivalent formulations of the Riemann Hypothesis. One such reformulation is in terms of multiplier sequences and another is in terms of sequences of hyperbolic polynomials.

In order to provide a bird's-eye view of the scope of this workshop, we highlight here, in general terms, some of the topics that were investigated.

♦ Young-One Kim and Haseo Ki reported new results concerning certain linear operators, the de Bruijn-Newman constant and the Riemann ξ- function;

♦ David Cardon promulgated an elegant geometric argument which shows that the coefficients of real entire functions whose zeros lie in a strip satisfy a certain concavity property enjoyed by hyperbolic polynomials;

♦ Julius Borcea and Mikhail Tyaglov (a doctoral student of Olga Holtz) proposed arguments to solve a 175-year-old open problem (now known as the "Hawaii problem") of Gauss (made precise in 1987). These talks generated considerable interest and led to invigorating discussions about the nature of certain level curves;

♦ Olga Holtz and Charles Johnson suggested some interesting unifying principles for classes of matrices that arose from the work of Pólya and Schur.

♦ At least eight participants (T. Craven, D. Dimitrov, S. Edwards, S. Fisk, O. Katkova, B. Shapiro, D. Cardon, A. Vishnyakova) shared ideas in an attempt to characterize multiplier sequences whose reciprocals are positive definite sequences.

♦ The Bessis-Moussa-Villani (BMV) conjecture pertaining to positive definite matrices was the theme of several group discussions;

♦ Web Director David Farmer, a superb lecturer, gave an informative talk about the movements of the zeros under differentiation and the Riemann Hypothesis;

In 1958 Peter Lax conjectured that hyperbolic polynomials in three variables are determinants of linear combinations of three symmetric matrices. This conjecture was recently proved by J. W. Helton and V. Vinnikov. Since multivariable hyperbolic polynomials have many surprising properties and since they are intimately linked with convex optimization and programming problems, there were a number stimulating group discussions pertaining to this circle of ideas.

The seminal work of Pólya and Schur has been the fount of numerous investigations and found important applications in the theory of integral transforms, approximation theory, the theory of total positivity of matrices and probability theory. The participants of this workshop discussed the interplay between specific problems related to the aforementioned applications. In a collaborative effort, in the course of small (and large) group discussions, experts or specialists, in fields such as, matrix theory, combinatorics, geometry, optimization theory, function theory in one or several variables, brought to light fresh approaches that may render certain problems tractable.

Modus Operandi

In sharp contrast to the format of conferences, this workshop provided a unique experience of interaction with researchers in diverse fields. The workshop venue, with its wonderful library and computing and other facilities, created a milieu which was conducive for the free exchange of ideas. Thanks to the assistance and the expert guidance of the Directors of AIM, the transition from a few, brief introductory talks led seamlessly to small (and large) group discussions dealing with problems of common interest. All the participants benefited from this dynamic collaboration.