Fekete polynomials, quadratic residues, and arithmetic

Zeros of Fekete Polynomials

Abelian groups and quadratic residues in weak arithmetic

We investigate the provability of some properties of abelian groups and quadratic residues in variants of bounded arithmetic. Specifically, we show that the structure theorem for finite abelian groups is provable in S 2 + iWPHP(Σ b 1), and use it to derive Fermat’s little theorem and Euler’s criterion for the Legendre symbol in S 2 + iWPHP(PV ) extended by the pigeonhole principle PHP(PV ). We ...

Fekete Polynomials and Shapes of Julia Sets

We prove that a nonempty, proper subset S of the complex plane can be approximated in a strong sense by polynomial filled Julia sets if and only if S is bounded and Ĉ \ int(S) is connected. The proof that such a set is approximable by filled Julia sets is constructive and relies on Fekete polynomials. Illustrative examples are presented. We also prove an estimate for the rate of approximation i...

Some Numerical Results on Fekete Polynomials

It is known that if x is a real residue character modulo k with x(p) = — 1 k n for the first five primes p, then the corresponding Fekete polynomial £„=j X(n)x changes sign on (0, 1). In this paper it is shown that the condition that x(p) be — 1 for the first four primes p is not sufficient to guarantee such a sign change. More specifically, if x ¡s the real nonprincipal character modulo either...

An Extremal Property of Fekete Polynomials