Newman Polynomials, Reducibility, and Roots on the Unit Circle
نویسنده
چکیده
A length k Newman polynomial is any polynomial of the form za1 + · · ·+zak (where a1 < · · · < ak). Some Newman polynomials are reducible over the rationals, and some are not. Some Newman polynomials have roots on the unit circle, and some do not. Defining, in a natural way, what we mean by the “proportion” of length k Newman polynomials with a given property, we prove that • 1/4 of length 3 Newman polynomials are reducible over the rationals • 1/4 of length 3 Newman polynomials have roots on the unit circle • 3/7 of length 4 Newman polynomials are reducible over the rationals • 3/7 of length 4 Newman polynomials have roots on the unit circle We also show that certain plausible conjectures imply that the proportion of length 5 Newman polynomials with roots on the unit circle is 909/9464.
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