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

DOI:
https://doi.org/10.1090/S0002-9939-00-05818-4

Published electronically:
November 30, 2000

MathSciNet review:
1823911

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Let be a monic polynomial of degree , with complex coefficients, and let be its monic factor. We prove an asymptotically sharp inequality of the form , where denotes the sup norm on a compact set in the plane. The best constant 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.

**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**

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

Email:
igor@math.okstate.edu

DOI:
https://doi.org/10.1090/S0002-9939-00-05818-4

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