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

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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]**T.M. Apostol,*Introduction to Analytic Number Theory*, Springer-Verlag, 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), 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]**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), 69-94. MR**25:1139**

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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