zbMATH — the first resource for mathematics

Pólya-Schur master theorems for circular domains and their boundaries. (English) Zbl 1184.30004
Let \(\Omega\subseteq \mathbb{C}\) be a closed circular domain (i.e., the image of the closed unit disk under a Möbius transformation) or its boundary. Denote by \(\pi(\Omega)\) the class of all (complex or real) univariate polynomials whose zeros lie in \(\Omega\), and by \(\pi_n(\Omega)\) the subclass of \(\pi(\Omega)\) consisting of polynomials of degree at most \( n\). In the paper, the following three problems are considered.
Problem 1. Characterize all linear transformations \( T:\pi(\Omega) \to \pi(\Omega)\cup \{0\}.\)
Problem 2. For \(n \in \mathbb{N}, \) describe all linear operators \( T:\pi_n(\Omega) \to \pi(\Omega)\cup \{0\}.\)
Problem 3. For \(n \in \mathbb{N}, \) describe all linear operators \( T:\pi_n(\Omega)\setminus \pi_{n-1}(\Omega) \to \pi(\Omega)\cup \{0\}.\)
The authors obtain exhaustive solutions of these problems. In particular, for \(\Omega = \{z \in \mathbb{C}: \operatorname{Im}(z)>0\}\), they give a description of all operators that preserve the property of polynomials to be stable. For \(\Omega =\mathbb{R}\), their results settle an open question on characterization of hyperbolicity preservers that goes back to Laguerre, Pólya, Schur. More precisely, they prove the following theorem.
Theorem. A linear operator \(T:\mathbb{R}[z] \to \mathbb{R}[z]\) is an operator of the kind \(T:\pi(\mathbb{R}) \to \pi(\mathbb{R})\cup \{0\}\) if and only if one of the following holds:
(a) \(T\) is of the form \(T(f)=\alpha (f)P+\beta (f)Q\), where \(\alpha, \beta: \mathbb{R}[z]\to \mathbb{R}\) are linear functionals, and \(P,Q \in \pi(\mathbb{R})\) have interlacing zeros;
(b) for all \(n \in \mathbb{N}\), either the polynomial \(T[(z+w)^n]\neq 0\) for \((z,w) \in \{(z,w) \in \mathbb{C}^2: \operatorname{Im} z >0,\operatorname{Im} w >0\}\) or \(T[(z+w)^n]\equiv 0\);
(c) for all \(n \in \mathbb{N}\), either the polynomial \(T[(z-w)^n]\neq 0\) for \((z,w) \in \{(z,w) \in \mathbb{C}^2: \operatorname{Im} z >0, \operatorname{Im} w >0\}\) or \(T[(z-w)^n]\equiv 0.\)

30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
26C10 Real polynomials: location of zeros
30D15 Special classes of entire functions of one complex variable and growth estimates
47D03 Groups and semigroups of linear operators
Full Text: DOI Link arXiv
[1] A. Aleman, D. Beliaev, and H. Hedenmalm, ”Real zero polynomials and Pólya-Schur type theorems,” J. Anal. Math., vol. 94, pp. 49-60, 2004. · Zbl 1081.30005
[2] M. F. Atiyah, R. Bott, and L. Gårding, ”Lacunas for hyperbolic differential operators with constant coefficients, I,” Acta Math., vol. 124, pp. 109-189, 1970. · Zbl 0191.11203
[3] J. Borcea and P. Brändén, ”Multivariate Polya-Schur classification problems in the Weyl algebra,” , Preprint , 2006. · Zbl 05624984
[4] J. Borcea and P. Brändén, ”Applications of stable polynomials to mixed determinants: Johnson’s conjectures, unimodality and symmetrized Fischer products,” Duke Math. J., vol. 143, iss. 2, pp. 205-223, 2008. · Zbl 1151.15013
[5] J. Borcea, P. Brändén, G. Csordas, and V. Vinnikov, ”Pólya-Schur-Lax problems: Hyperbolicity and stability preservers,” , workshop, American Institute of Mathematics, May-June 2007, http://www.aimath.org , 2007.
[6] P. Brändén, ”Polynomials with the half-plane property and matroid theory,” Adv. Math., vol. 216, iss. 1, pp. 302-320, 2007. · Zbl 1128.05014
[7] P. Brändén, ”On linear transformations preserving the Pólya frequency property,” Transactions Amer. Math. Soc., vol. 358, iss. 8, pp. 3697-3716, 2006. · Zbl 1086.05007
[8] F. Brenti, ”Unimodal, log-concave and Pólya frequency sequences in combinatorics,” Memoirs Amer. Math. Soc., vol. 81, iss. 413, p. viii, 1989. · Zbl 0697.05011
[9] N. G. de Bruijn, ”The roots of trigonometric integrals,” Duke Math. J., vol. 17, pp. 197-226, 1950. · Zbl 0038.23302
[10] J. M. Carnicer, J. M. Peña, and A. Pinkus, ”On some zero-increasing operators,” Acta Math. Hungar., vol. 94, iss. 3, pp. 173-190, 2002. · Zbl 0997.26008
[11] J. M. Carnicer, J. M. Peña, and A. Pinkus, ”On zero-preserving linear transformations,” J. Math. Anal. Appl., vol. 266, iss. 1, pp. 237-258, 2002. · Zbl 0995.41015
[12] T. Craven and G. Csordas, ”Composition theorems, multiplier sequences and complex zero decreasing sequences,” in Value Distribution Theory and Related Topics, Barsegian, G., Laine, I., and Yang, C. C., Eds., Boston: Kluwer, 2004, pp. 131-166. · Zbl 1101.26015
[13] T. Craven and G. Csordas, ”Multiplier sequences for fields,” Illinois J. Math., vol. 21, iss. 4, pp. 801-817, 1977. · Zbl 0389.12009
[14] T. Craven and G. Csordas, ”The Gauss-Lucas theorem and Jensen polynomials,” Transactions Amer. Math. Soc., vol. 278, iss. 1, pp. 415-429, 1983. · Zbl 0497.30022
[15] T. Craven, G. Csordas, and W. Smith, ”The zeros of derivatives of entire functions and the Pólya-Wiman conjecture,” Ann. of Math., vol. 125, iss. 2, pp. 405-431, 1987. · Zbl 0625.30036
[16] G. Csordas, ”Linear operators and the distribution of zeros of entire functions,” Complex Variables Elliptic Equ., vol. 51, iss. 7, pp. 625-632, 2006. · Zbl 1104.30017
[17] S. Fisk, ”Polynomials, roots, and interlacing,” 2006.
[18] J. H. Grace, ”On the zeros of a polynomial,” Proc. Cambridge Philos. Soc., vol. 11, pp. 352-356, 1902. · JFM 33.0121.04
[19] G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, 2nd ed., Cambridge University Press, 1988. · Zbl 0634.26008
[20] S. Hellerstein and J. Rossi, ”On the distribution of zeros of solutions of second-order differential equations,” Complex Variables Theory Appl., vol. 13, iss. 1-2, pp. 99-109, 1989. · Zbl 0688.30023
[21] E. Hille, Lectures on Ordinary Differential Equations, Addison-Wesley Publ. Co., 1969. · Zbl 0179.40301
[22] A. Iserles, S. P. Nørsett, and E. B. Saff, ”On transformations and zeros of polynomials,” Rocky Mountain J. Math., vol. 21, iss. 1, pp. 331-357, 1991. · Zbl 0754.26007
[23] A. Iserles and S. P. Nørsett, ”Bi-orthogonality and zeros of transformed polynomials,” Jour. Comput. Appl. Math., vol. 19, iss. 1, pp. 39-45, 1987. · Zbl 0636.42022
[24] A. Iserles and S. P. Nørsett, ”Zeros of transformed polynomials,” SIAM J. Math. Anal., vol. 21, iss. 2, pp. 483-509, 1990. · Zbl 0698.33007
[25] A. Iserles and E. B. Saff, ”Zeros of expansions in orthogonal polynomials,” Math. Proc. Cambridge Philos. Soc., vol. 105, iss. 3, pp. 559-573, 1989. · Zbl 0679.33007
[26] E. Laguerre, ”Fonctions du genre zéro et du genre un,” C. R. Acad. Sci. Paris, vol. 95, pp. 828-831, 1882. · JFM 14.0057.05
[27] T. D. Lee and C. N. Yang, ”Statistical theory of equations of state and phase transitions, II: Lattice gas and Ising model,” Physical Rev., vol. 87, pp. 410-419, 1952. · Zbl 0048.43401
[28] J. B. Levin, Distribution of Zeros of Entire Functions, Revised ed., Providence, R.I.: Amer. Math. Soc., 1980.
[29] E. Lindwart and G. Pólya, ”Über einen Zusammenhang zwischen der Konvergenz von Polynomfolgen und der Verteilung ihrer Wurzeln,” Rend. Circ. Mat. Palermo, vol. 37, pp. 297-304, 1914. · JFM 45.0650.01
[30] M. Marden, The Geometry of the Zeros of a Polynomial in a Complex Variable, New York: American Mathematical Society, 1949. · Zbl 0038.15303
[31] N. Obreschkoff, ”Sur les fonctions entières limites de polynômes dont les zéros sont réel et entrelacés,” Mat. Sbornik, vol. 9 (51), pp. 421-428, 1941. · Zbl 0024.32903
[32] N. Obreschkoff, Verteilung und Berechnung der Nullstellen Reeller Polynome, Berlin: VEB Deutscher Verlag der Wissenschaften, 1963. · Zbl 0156.28202
[33] A. Pinkus, ”Some remarks on zero-increasing transformations,” in Approximation Theory, X: Abstract and Classical Analysis, Chui, C. K., Schumaker, L. L., and Stoeckler, J., Eds., Nashville: Vanderbilt Univ. Press, 2002, pp. 333-352. · Zbl 1061.30006
[34] G. Pólya, ”Bemerkung Über die Integraldarstellung der Riemannschen \(\zeta\)-Funktion,” Acta Math., vol. 48, iss. 3-4, pp. 305-317, 1926. · JFM 52.0335.02
[35] G. Pólya, ”Über Annäherung durch Polynome mit lauter reellen Wurzeln,” Rend. Circ. Mat. Palermo, vol. 36, pp. 279-295, 1913. · JFM 44.0474.01
[36] G. Pólya and I. Schur, ”Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen,” J. Reine Angew. Math., vol. 144, pp. 89-113, 1914. · JFM 45.0176.01
[37] G. Pólya and G. Szegö, Problems and Theorems in Analysis, II: Theory of Functions, Zeros, Polynomials, Determinants, Number Theory, Geometry, New York: Springer, 1976. · Zbl 0359.00003
[38] Q. I. Rahman and G. Schmeisser, Analytic Theory of Polynomials, Oxford: Oxford University Press, 2002. · Zbl 1072.30006
[39] I. Schur, ”Zwei Sätze über algebraische Gleichungen mit lauter reellen Wurzeln,” J. Reine Angew. Math., vol. 144, pp. 75-88, 1923. · JFM 45.0169.01
[40] I. Schur, ”Über eine Klasse von Mittelbildungen mit Anwendungen auf die Determinantentheorie,” Sitzungsber. Berl. Math. Ges., vol. 22, pp. 9-20, 1923. · JFM 49.0054.01
[41] O. Szász, ”On sequences of polynomials and the distribution of their zeros,” Bull. Amer. Math. Soc., vol. 49, pp. 377-383, 1943. · Zbl 0060.05304
[42] G. Szegö, ”Bemerkungen zu einem Satz von J. H. Grace über die Wurzeln algebraischer Gleichungen,” Math. Z., vol. 13, iss. 1, pp. 28-55, 1922. · JFM 48.0082.02
[43] J. L. Walsh, ”On the location of the roots of certain types of polynomials,” Trans. Amer. Math. Soc., vol. 24, iss. 3, pp. 163-180, 1922.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.