On the span of polynomials with integer coefficients. (English) Zbl 1216.12001

Summary: Following a paper of R. M. Robinson [Math. Comput. 18, 547–559 (1964; Zbl 0147.12905)], we classify all hyperbolic polynomials in one variable with integer coefficients and span less than 4 up to degree 14, and with some additional hypotheses, up to degree 17. We conjecture that the classification is also complete for degrees 15, 16, and 17.
Besides improving on the method used by Robinson, we develop new techniques that turn out to be of some interest.
A close inspection of the polynomials thus obtained shows some properties deserving further investigations.


12D10 Polynomials in real and complex fields: location of zeros (algebraic theorems)
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
11C08 Polynomials in number theory


Zbl 0147.12905
Full Text: DOI


[1] Eberhard Becker and Thorsten Wöermann, On the trace formula for quadratic forms, Recent advances in real algebraic geometry and quadratic forms (Berkeley, CA, 1990/1991; San Francisco, CA, 1991) Contemp. Math., vol. 155, Amer. Math. Soc., Providence, RI, 1994, pp. 271 – 291. · Zbl 0835.11016
[2] C. Hermite, Remarques sur le théorème de Sturm, C. R. Acad. Sci. Paris, 36 (1853), 32-54.
[3] L. Kronecker. Zwei Sätze über Gleichungen mit ganzzahligen Coefficienten, J. Reine Angew. Math., 53 (1857), 173-175. · ERAM 053.1389cj
[4] P. Pedersen, M.-F. Roy, and A. Szpirglas, Counting real zeros in the multivariate case, Computational algebraic geometry (Nice, 1992) Progr. Math., vol. 109, Birkhäuser Boston, Boston, MA, 1993, pp. 203 – 224. · Zbl 0806.14042
[5] Q. I. Rahman and G. Schmeisser, Analytic theory of polynomials, London Mathematical Society Monographs. New Series, vol. 26, The Clarendon Press, Oxford University Press, Oxford, 2002. · Zbl 1072.30006
[6] J. C. Mason and D. C. Handscomb, Chebyshev polynomials, Chapman & Hall/CRC, Boca Raton, FL, 2003. · Zbl 1015.33001
[7] Raphael M. Robinson, Intervals containing infinitely many sets of conjugate algebraic integers, Studies in mathematical analysis and related topics, Stanford Univ. Press, Stanford, Calif., 1962, pp. 305 – 315.
[8] Raphael M. Robinson, Algebraic equations with span less than 4, Math. Comp. 18 (1964), 547 – 559. · Zbl 0147.12905
[9] I. Schur, Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten, Math. Z. 1 (1918), no. 4, 377 – 402 (German). · JFM 46.0128.03
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.