Abstract
We say that a polynomial p of degree n is self-reciprocal polynomial if \(p(z) = {z}^{n}p(1/z)\), i.e., if its coefficients are “symmetric.” This chapter surveys the literature on zeros of this family of complex polynomials, with the focus on criteria determining when such polynomials have all their roots on the unit circle. The last section contains a new conjectural criteria which, if true, would have very interesting applications.
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Notes
- 1.
Some authors, such as Chen [2], have p ∗ denote the complex conjugate of the reverse polynomial. It will not matter for us, since we will eventually assume that the coefficients are real.
- 2.
These terms will not be defined precisely here. Please see standard texts for a rigorous treatment.
- 3.
Sometimes called the reciprocal of the Frobenius polynomial, or the zeta polynomial.
- 4.
In this definition, we assume for simplicity a 2n ≠ 0; see [12] for the general definition of p↦T p .
- 5.
In fact, both are exercises in Marden [14].
- 6.
In one sense, Chinen’s version is slightly stronger, and it is that version which we are stating.
- 7.
For example, numerical experiments suggest “linear growth” seems too fast but “logarithmic growth” seems sufficient.
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Joyner, D. (2013). Zeros of Some Self-Reciprocal Polynomials. In: Andrews, T., Balan, R., Benedetto, J., Czaja, W., Okoudjou, K. (eds) Excursions in Harmonic Analysis, Volume 1. Applied and Numerical Harmonic Analysis. Birkhäuser, Boston. https://doi.org/10.1007/978-0-8176-8376-4_17
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