Papers › Fekete polynomials of principal Dirichlet characters
Fekete polynomials of principal Dirichlet characters
Shiva Chidambaram, Ján Mináč, Tung T. Nguyen, Nguyen Duy Tân
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Fekete polynomials associated to quadratic Dirichlet characters have interesting arithmetic properties, and have been studied in many works. In this paper, we study a seemingly simpler yet rich variant: the Fekete polynomial Fₙ(x) = ∑ₐ₌₁ⁿ χₙ(a) xᵃ associated to a principal Dirichlet character χₙ of modulus n. We investigate the cyclotomic factors of Fₙ and conjecturally describe all of them. One interesting observation from our computations is that the non-cyclotomic part fₙ of Fₙ(x)/x seems to be always irreducible. We study this factor closely in the special case that n is a product of two odd primes, proving separability in specific cases, and studying its coefficients and special values. Combining these theoretical results with computational evidence lets us identify the Galois group of fₙ for small n, and raises precise questions in general.
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