Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

On the span of polynomials with integer coefficients


Authors: Stefano Capparelli, Alberto Del Fra and Carlo Sciò
Journal: Math. Comp. 79 (2010), 967-981
MSC (2000): Primary 12D10; Secondary 30C15, 11C08
DOI: https://doi.org/10.1090/S0025-5718-09-02292-3
Published electronically: September 8, 2009
MathSciNet review: 2600551
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 1. E. Becker and T. Wörmann, On the trace formula for quadratic forms, Contemp. Math., 155 (1994), 271-291. MR 1260713 (95f:12002)
  • 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.
  • 4. P. Pedersen, M. F. Roy and A. Szpirglas, Counting real zeros in the multivariate case, Progr. Math., 109 (1993), 203-223. MR 1230868 (94m:14075)
  • 5. Q. I. Rahman and G. Schmeisser, Analytic Theory of Polynomials, London Mathematical Society New Series, 26, Oxford Univ. Press 2002. MR 1954841 (2004b:30015)
  • 6. J. C. Mason and D. C. Handscomb, Chebyshev polynomials. Chapman & Hall/CRC, Boca Raton, 2003. MR 1937591 (2004h:33001)
  • 7. R. Robinson, Intervals containing infinitely many sets of conjugate algebraic integers, Studies in Mathematical Analysis and Related Topics: Essays in honor of George Pólya, Stanford Univ. Press, 1962, 305-315. MR 0144892 (26:2433)
  • 8. R. Robinson, Algebraic equations with span less than 4, Math. Comp., 18 (1964), 547-559. MR 0169374 (29:6624)
  • 9. I. Schur, Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten, Math. Z., 1 (1918), 377-402. MR 1544303

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 12D10, 30C15, 11C08

Retrieve articles in all journals with MSC (2000): 12D10, 30C15, 11C08


Additional Information

Stefano Capparelli
Affiliation: Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate, Università di Roma “La Sapienza”, Via Scarpa 16, I-00161 Roma, Italy
Email: capparelli@dmmm.uniroma1.it

Alberto Del Fra
Affiliation: Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate, Università di Roma “La Sapienza”, Via Scarpa 16, I-00161 Roma, Italy
Email: alberto.delfra@uniroma1.it

Carlo Sciò
Affiliation: ENEA FIM, Via E. Fermi 45, I-00044 Frascati (RM), Italy
Email: scio@frascati.enea.it

DOI: https://doi.org/10.1090/S0025-5718-09-02292-3
Received by editor(s): November 10, 2008
Received by editor(s) in revised form: April 4, 2009
Published electronically: September 8, 2009
Article copyright: © Copyright 2009 American Mathematical Society

American Mathematical Society