On the span of polynomials with integer coefficients

Capparelli, Stefano; Del Fra, Alberto; Sciò, Carlo

doi:10.1090/S0025-5718-09-02292-3

On the span of polynomials with integer coefficients
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by Stefano Capparelli, Alberto Del Fra and Carlo Sciò PDF

Math. Comp. 79 (2010), 967-981 Request permission

Abstract:

Following a paper of R. Robinson, 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.

References

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. MR 1260713, DOI 10.1090/conm/155/01385
C. Hermite, Remarques sur le théorème de Sturm, C. R. Acad. Sci. Paris, 36 (1853), 32-54.
L. Kronecker. Zwei Sätze über Gleichungen mit ganzzahligen Coefficienten, J. Reine Angew. Math., 53 (1857), 173-175.
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. MR 1230868, DOI 10.1007/978-1-4612-2752-6_{1}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. MR 1954841
J. C. Mason and D. C. Handscomb, Chebyshev polynomials, Chapman & Hall/CRC, Boca Raton, FL, 2003. MR 1937591
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. MR 0144892
Raphael M. Robinson, Algebraic equations with span less than $4$, Math. Comp. 18 (1964), 547–559. MR 169374, DOI 10.1090/S0025-5718-1964-0169374-X
I. Schur, Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten, Math. Z. 1 (1918), no. 4, 377–402 (German). MR 1544303, DOI 10.1007/BF01465096