On a Problem of Byrnes concerning Polynomials with Restricted Coefficients

Author:
David W. Boyd

Journal:
Math. Comp. **66** (1997), 1697-1703

MSC (1991):
Primary 11C08, 12D10; Secondary 94A99, 11Y99

DOI:
https://doi.org/10.1090/S0025-5718-97-00892-2

MathSciNet review:
1433263

Full-text 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*, Springer-Verlag, New York-Heidelberg, 1976. Undergraduate Texts in Mathematics. MR**0434929****[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), 301-303.**[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. 677-678.**[5]**M. Mignotte,*Sur les polynômes divisibles par*, Arithmetix**2**(1980), 28-29.**[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****[7]**J. Barkley Rosser and Lowell Schoenfeld,*Approximate formulas for some functions of prime numbers*, Illinois J. Math.**6**(1962), 64–94. MR**0137689**

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:
https://doi.org/10.1090/S0025-5718-97-00892-2

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