Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



An inequality for the norm of a polynomial factor

Author: Igor E. Pritsker
Journal: Proc. Amer. Math. Soc. 129 (2001), 2283-2291
MSC (1991): Primary 30C10, 30C85; Secondary 11C08, 31A15
Published electronically: November 30, 2000
MathSciNet review: 1823911
Full-text PDF

Abstract | References | Similar Articles | Additional Information


Let $p(z)$ be a monic polynomial of degree $n$, with complex coefficients, and let $q(z)$ be its monic factor. We prove an asymptotically sharp inequality of the form $\Vert q\Vert _{E} \le C^n \, \Vert p\Vert _E$, where $\Vert\cdot\Vert _E$ denotes the sup norm on a compact set $E$ in the plane. The best constant $C_E$ in this inequality is found by potential theoretic methods. We also consider applications of the general result to the cases of a disk and a segment.

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

  • 1. P. B. Borwein, Exact inequalities for the norms of factors of polynomials, Can. J. Math. 46 (1994), 687-698. MR 95k:26015
  • 2. P. Borwein and T. Erdélyi, Polynomials and Polynomial Inequalities, Springer-Verlag, New York, 1995. MR 97e:41001
  • 3. D. W. Boyd, Two sharp inequalities for the norm of a factor of a polynomial, Mathematika 39 (1992), 341-349. MR 94a:11162
  • 4. D. W. Boyd, Sharp inequalities for the product of polynomials, Bull. London Math. Soc. 26 (1994), 449-454. MR 95m:30008
  • 5. D. W. Boyd, Large factors of small polynomials, Contemp. Math. 166 (1994), 301-308. MR 95e:11034
  • 6. P. Glesser, Nouvelle majoration de la norme des facteurs d'un polynôme, C. R. Math. Rep. Acad. Sci. Canada 12 (1990), 224-228. MR 92d:12002
  • 7. A. Granville, Bounding the coefficients of a divisor of a given polynomial, Monatsh. Math. 109 (1990), 271-277. MR 91g:12001
  • 8. S. Landau, Factoring polynomials quickly, Notices Amer. Math. Soc. 34 (1987), 3-8. MR 87k:12002
  • 9. M. Mignotte, Some useful bounds, In ``Computer Algebra, Symbolic and Algebraic Computation" (B. Buchberger et al., eds.), pp. 259-263, Springer-Verlag, New York, 1982. CMP 16:06
  • 10. I. E. Pritsker, Products of polynomials in uniform norms, to appear in Trans. Amer. Math. Soc.
  • 11. I. E. Pritsker, Comparing norms of polynomials in one and several variables, J. Math. Anal. Appl. 216 (1997), 685-695. MR 93j:30002
  • 12. T. Ransford, Potential Theory in the Complex Plane, Cambridge University Press, Cambridge, 1995. MR 96e:31001
  • 13. M. Tsuji, Potential Theory in Modern Function Theory, Chelsea Publ. Co., New York, 1975. MR 54:2990

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 30C10, 30C85, 11C08, 31A15

Retrieve articles in all journals with MSC (1991): 30C10, 30C85, 11C08, 31A15

Additional Information

Igor E. Pritsker
Affiliation: Department of Mathematics, 401 Mathematical Sciences, Oklahoma State University, Stillwater, Oklahoma 74078-1058

Keywords: Polynomials, uniform norm, logarithmic capacity, equilibrium measure, subharmonic function, Fekete points
Received by editor(s): November 8, 1999
Received by editor(s) in revised form: November 23, 1999
Published electronically: November 30, 2000
Additional Notes: Research supported in part by the National Science Foundation grants DMS-9996410 and DMS-9707359.
Communicated by: Albert Baernstein II
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society