On a Problem of Byrnes concerning Polynomials with Restricted Coefficients
Author:
David W. Boyd
Journal:
Math. Comp. 66 (1997), 16971703
MSC (1991):
Primary 11C08, 12D10; Secondary 94A99, 11Y99
MathSciNet review:
1433263
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: We consider a question of Byrnes concerning the minimal degree of a polynomial with all coefficients in which has a zero of a given order at . For , we prove his conjecture that the monic polynomial of this type of minimal degree is given by , but we disprove this for . We prove that a polynomial of this type must have , which is in sharp contrast with the situation when one allows coefficients in . The proofs use simple number theoretic ideas and depend ultimately on the fact that .
 [1]
Tom
M. Apostol, Introduction to analytic number theory,
SpringerVerlag, New YorkHeidelberg, 1976. Undergraduate Texts in
Mathematics. MR
0434929 (55 #7892)
 [2]
P. Borwein, T. Erdélyi & G. Kós, Polynomials with Restricted Coefficients (to appear).
 [3]
J.S. Byrnes & D.J. Newman, Null Steering Employing Polynomials with Restricted Coefficients, IEEE Trans. Antennas and Propagation 36 (1988), 301303.
 [4]
J.S. Byrnes, Problems on Polynomials with Restricted Coefficients Arising from Questions in Antenna Array Theory, Recent Advances in Fourier Analysis and Its Applications (J.S. Byrnes & J.F. Byrnes, eds.), Kluwer Academic Publishers, Dordrecht, 1990, pp. 677678.
 [5]
M. Mignotte, Sur les polynômes divisibles par , Arithmetix 2 (1980), 2829.
 [6]
Aleksej
D. Korshunov, On the number of nonisomorphic strongly connected
finite automata, Elektron. Informationsverarb. Kybernet.
22 (1986), no. 9, 459–462 (English, with German
and Russian summaries). MR 862029
(88a:68076)
 [7]
J.
Barkley Rosser and Lowell
Schoenfeld, Approximate formulas for some functions of prime
numbers, Illinois J. Math. 6 (1962), 64–94. MR 0137689
(25 #1139)
 [1]
 T.M. Apostol, Introduction to Analytic Number Theory, SpringerVerlag, Berlin and New York, 1976. MR 55:7892
 [2]
 P. Borwein, T. Erdélyi & G. Kós, Polynomials with Restricted Coefficients (to appear).
 [3]
 J.S. Byrnes & D.J. Newman, Null Steering Employing Polynomials with Restricted Coefficients, IEEE Trans. Antennas and Propagation 36 (1988), 301303.
 [4]
 J.S. Byrnes, Problems on Polynomials with Restricted Coefficients Arising from Questions in Antenna Array Theory, Recent Advances in Fourier Analysis and Its Applications (J.S. Byrnes & J.F. Byrnes, eds.), Kluwer Academic Publishers, Dordrecht, 1990, pp. 677678.
 [5]
 M. Mignotte, Sur les polynômes divisibles par , Arithmetix 2 (1980), 2829.
 [6]
 A. Nijenhuis & H.S. Wilf, Combinatorial Algorithms, Academic Press, Orlando, 1978. MR 88a:68076
 [7]
 J.B. Rosser & L. Schoenfeld, Approximate formulas for some functions of prime number theory, Illinois J. Math 6 (1962), 6994. MR 25:1139
Similar Articles
Retrieve articles in Mathematics of Computation of the American Mathematical Society
with MSC (1991):
11C08,
12D10,
94A99,
11Y99
Retrieve articles in all journals
with MSC (1991):
11C08,
12D10,
94A99,
11Y99
Additional Information
David W. Boyd
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, B.C., Canada V6T 1Z2
Email:
boyd@math.ubc.ca
DOI:
http://dx.doi.org/10.1090/S0025571897008922
PII:
S 00255718(97)008922
Keywords:
Polynomial,
zero,
antenna array,
notch filter
Received by editor(s):
November 16, 1995
Additional Notes:
This research was supported by a grant from NSERC
Article copyright:
© Copyright 1997
American Mathematical Society
