Fekete Polynomials Research Articles

On the Arithmetic of Generalized Fekete Polynomials

For each prime number p one can associate a Fekete polynomial with coefficients–1 or 1 except the constant term, which is 0. These are classical polynomials that have been studied extensively in the framework of analytic number theory. In a recent paper, we showed that these polynomials also encode interesting arithmetic information. In this article, we define generalized Fekete polynomials associated with quadratic characters whose conductors could be composite numbers. We then investigate the appearance of cyclotomic factors in these generalized Fekete polynomials. Based on this investigation, we introduce a compact version of Fekete polynomials as well as their trace polynomials. We then study the Galois groups of these Fekete polynomials using modular techniques. In particular, we discover some surprising extra symmetries which imply some restrictions on the corresponding Galois groups. Finally, based on both theoretical and numerical data, we propose a precise conjecture on the structure of these Galois groups.

Fekete polynomials, quadratic residues, and arithmetic

Fekete polynomials associate with each prime number p a polynomial with coefficients −1 or 1 except the constant term, which is 0. These coefficients reflect the distribution of quadratic residues modulo p. These polynomials were already considered in the 19th century in relation to the studies of Dirichlet L-functions. In our paper, we introduce two closely related polynomials. We then express their special values at several integers in terms of certain class numbers and generalized Bernoulli numbers. Additionally, we study the splitting fields and the Galois group of these polynomials. In particular, we propose two conjectures on the structure of these Galois groups. We also provide some computational evidence toward the validity of these conjectures.

A necessary and sufficient condition for convergence of the zeros of random polynomials

Consider random polynomials of the form Gn=∑i=0nξipi, where the ξi are i.i.d. non-degenerate complex random variables, and {pi} is a sequence of orthonormal polynomials with respect to a regular measure τ supported on a compact set K. We show that the zero measure of Gn converges weakly almost surely to the equilibrium measure of K if and only if Elog⁡(1+|ξ0|)<∞. This generalizes the corresponding result of Ibragimov and Zaporozhets in the case when pi(z)=zi. We also show that the zero measure of Gn converges weakly in probability to the equilibrium measure of K if and only if P(|ξ0|>en)=o(n−1).Our proofs rely on results from small ball probability and exploit the structure of general orthogonal polynomials. Our methods also work for sequences of asymptotically minimal polynomials in Lp(τ), where p∈(0,∞]. In particular, sequences of Lp-minimal polynomials and (normalized) Faber and Fekete polynomials fall into this class.

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 $\hat{\mathbb{C}} \setminus \textrm{int}(S)$ 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 in terms of geometric and potential theoretic quantities.

Improved Lower Bound for the Mahler Measure of the Fekete Polynomials

We show that there is an absolute constant $$c > 1/2$$ such that the Mahler measure of the Fekete polynomials $$f_p$$ of the form $$\begin{aligned} f_p(z) := \sum _{k=1}^{p-1}{\left( \frac{k}{p} \right) z^k} \end{aligned}$$ (where the coefficients are the usual Legendre symbols) is at least $$c\sqrt{p}$$ for all sufficiently large primes p. This improves the lower bound $$\left( \frac{1}{2} - \varepsilon \right) \sqrt{p}$$ known before for the Mahler measure of the Fekete polynomials $$f_p$$ for all sufficiently large primes $$p \ge c_{\varepsilon }$$ . Our approach is based on the study of the zeros of the Fekete polynomials on the unit circle.

Lq Norms of Fekete and Related Polynomials

AbstractA Littlewood polynomial is a polynomial in ℂ[z] having all of its coefficients in {−1, 1}. There are various old unsolved problems, mostly due to Littlewood and Erdos, that ask for Littlewood polynomials that provide a good approximation to a function that is constant on the complex unit circle, and in particular have small Lq normon the complex unit circle. We consider the Fekete polynomialswhere p is an odd prime and (· |p) is the Legendre symbol (so that z-1fp(z) is a Littlewood polynomial). We give explicit and recursive formulas for the limit of the ratio of Lq and L2 norm of fp when q is an even positive integer and p → ∞. To our knowledge, these are the first results that give these limiting values for specific sequences of nontrivial Littlewood polynomials and infinitely many q. Similar results are given for polynomials obtained by cyclically permuting the coefficients of Fekete polynomials and for Littlewood polynomials whose coefficients are obtained from additive characters of finite fields. These results vastly generalise earlier results on the L4 norm of these polynomials.

Power series approximations to Fekete polynomials

We study how well Fekete polynomials Fp(X)=∑n=0p−1npXn∈Z[X]with the coefficients given by Legendre symbols modulo a prime p can be approximated by power series representing algebraic functions of a given degree. We also obtain some explicit results describing polynomial recurrence relations which are satisfied by the coefficients of such algebraic functions.

On the sparsity of positive-definite automorphic forms within a family

Baker and Montgomery have proved that almost all Fekete polynomials with respect to a certain ordering have at least one zero on the interval (0, 1). It is also known that a Fekete polynomial has no zeros on the interval (0, 1) if and only if the corresponding automorphic form is positive-definite. Generalizing their result, we formulate an axiomatic result about sets of automorphic forms π satisfying certain averages when suitably ordered to ensure that almost all p’s are not positive-definite within such sets. We then apply the result to various families, including the family of holomorphic cusp forms, the family of Hilbert class characters of imaginary quadratic fields, and the family of elliptic curves. In the appendix, we apply the result to general families of automorphic forms defined by Sarnak, Shin, and Templer.

Littlewood polynomials with small [formula omitted] norm

Littlewood asked how small the ratio ‖f‖4/‖f‖2 (where ‖⋅‖α denotes the Lα norm on the unit circle) can be for polynomials f having all coefficients in {1,−1}, as the degree tends to infinity. Since 1988, the least known asymptotic value of this ratio has been 7/64, which was conjectured to be minimum. We disprove this conjecture by showing that there is a sequence of such polynomials, derived from the Fekete polynomials, for which the limit of this ratio is less than 22/194.

Upper bounds for the Lqnorm of Fekete polynomials on subarcs

Upper bounds for the L_qnorm of Fekete polynomials on subarcs

Sieve-Type Lower Bounds for the Mahler Measure of Polynomials on Subarcs

We prove sieve-type lower bounds for the Mahler measure of polynomials on subarcs of the unit circle of the complex plane. This is then applied to give an essentially sharp lower bound for the Mahler measure of the Fekete polynomials on subarcs.

Large Sieve Inequalities via Subharmonic Methods and the Mahler Measure of the Fekete Polynomials

AbstractWe investigate large sieve inequalities such as where is convex and increasing, P is a polynomial or an exponential of a potential, and the constant C depends on the degree of P, and the distribution of the points 0 ≤ Ƭ1 &lt; Ƭ2 &lt; … &lt; Ƭm ≤ 2π. The method allows greater generality and is in some ways simpler than earlier ones. We apply our results to estimate the Mahler measure of Fekete polynomials.

Explicit merit factor formulae for Fekete and Turyn polynomials

We give explicit formulas for the L 4 L_{4} norm (or equivalently for the merit factors) of various sequences of polynomials related to the Fekete polynomials \[ f q ( z ) := ∑ k = 1 q − 1 ( k q ) z k f_{q}(z) := \sum ^{q-1}_{k=1} \left (\frac {k}{q}\right ) z^{k} \] where ( ⋅ q ) \left (\frac {\cdot }{q}\right ) is the Legendre symbol. For example for q q an odd prime, \[ ‖ f q ‖ 4 4 := 5 q 2 3 − 3 q + 4 3 − 12 ( h ( − q ) ) 2 \|f_{q}\|_{4}^{4} : = \frac {5q^{2}}{3}-3q+ \frac {4}{3} - 12 (h(-q))^{2} \] where h ( − q ) h(-q) is the class number of Q ( − q ) \mathbb {Q}(\sqrt {-q}) . Similar explicit formulas are given for various polynomials including an example of Turyn’s that is constructed by cyclically permuting the first quarter of the coefficients of f q f_{q} . This is the sequence that has the largest known asymptotic merit factor. Explicitly, \[ R q ( z ) := ∑ k = 0 q − 1 ( k + [ q / 4 ] q ) z k R_{q}(z) := \sum ^{q-1}_{k=0} \left (\frac {k+[q/4] }{q}\right ) z^{k} \] where [ ⋅ ] [\cdot ] denotes the nearest integer, satisfies \[ ‖ R q ‖ 4 4 = 7 q 2 6 − q − 1 6 − γ q \|R_{q}\|_{4}^{4} = \frac {7q^{2}}{6}- {q} - \frac {1}{6} - \gamma _{q} \] where \[ γ q := { h ( − q ) ( h ( − q ) − 4 ) a m p ; if q ≡ 1 , 5 ( mod 8 ) , 12 ( h ( − q ) ) 2 a m p ; if q ≡ 3 ( mod 8 ) , 0 a m p ; if q ≡ 7 ( mod 8 ) . \gamma _{q}: = \begin {cases} h(-q) (h(-q)-4) &amp; \text {if $q \equiv 1,5 \pmod 8$},\\ 12 (h(-q))^{2} &amp; \text {if $q \equiv 3 \pmod 8$}, \\ 0 &amp; \text {if $q \equiv 7 \pmod 8$}. \end {cases} \] Indeed we derive a closed form for the L 4 L_{4} norm of all shifted Fekete polynomials \[ f q t ( z ) := ∑ k = 0 q − 1 ( k + t q ) z k . f_{q}^{t}(z) := \sum ^{q-1}_{k=0} \left (\frac {k+t}{q}\right ) z^{k}. \] Namely ‖ f q t ‖ 4 4 a m p ; = 1 3 ( 5 q 2 + 3 q + 4 ) + 8 t 2 − 4 q t − 8 t a m p ; − 8 q 2 ( 1 − 1 2 ( − 1 q ) ) | ∑ n = 1 q − 1 n ( n + t q ) | 2 , \begin{align*} \| f_{q}^{t} \|_{4}^{4} &amp;= \frac {1}{3}(5q^{2}+3q+4)+8t^{2}-4qt-8t &amp;\quad -\frac {8}{q^{2}}\left ( 1-\frac {1}{2} \left (\frac {-1}{q}\right ) \right ) \left |{\displaystyle \sum _{n=1}^{q-1}n \left (\frac {n+t}{q}\right )} \right |^{2}, \end{align*} and ‖ f q q − t + 1 ‖ 4 4 = ‖ f q t ‖ 4 4 \| f_{q}^{q-t+1} \|_{4}^{4}= \| f_{q}^{t} \|_{4}^{4} if 1 ≤ t ≤ ( q + 1 ) / 2 1 \le t \le (q+1)/2 .

Fekete Potentials and Polynomials for Continua

For planar continua, upper and lower bounds are given for the growth of the associated Fekete potentials, polynomials and energies. The main result is that for continua K of capacity 1 whose outer boundary is an analytic Jordan curve, the family of Fekete polynomials is bounded on K . Our work makes use of precise results of Pommerenke on the growth of the discriminant and on the distribution of the Fekete points. We also use potential theory, including the exterior Green function with pole at infinity. The Lipschitz character of this function determines the separation of the Fekete points.

An extremal property of Fekete polynomials

The Fekete polynomials are defined as \[ F_q(z) := \sum ^{q-1}_{k=1} \left (\frac {k}{q}\right ) z^k\] where $\left (\frac {\cdot }{q}\right )$ is the Legendre symbol. These polynomials arise in a number of contexts in analysis and number theory. For example, after cyclic permutation they provide sequences with smallest known $L_4$ norm out of the polynomials with $\pm 1$ coefficients. The main purpose of this paper is to prove the following extremal property that characterizes the Fekete polynomials by their size at roots of unity. Theorem 0.1. Let $f(x)=a_1x+a_2x^2+\cdots +a_{N-1}x^{N-1}$ with odd $N$ and $a_n=\pm 1$. If \[ \operatorname {max}\{ |f(\omega ^k)| : 0 \le k \le N-1 \} = \sqrt {N}, \] then $N$ must be an odd prime and $f(x)$ is $\pm F_q(x)$. Here $\omega :=e^{\frac {2\pi i}{N}}.$ This result also gives a partial answer to a problem of Harvey Cohn on character sums.

Some Numerical Results on Fekete Polynomials

It is known that if $\chi$ is a real residue character modulo k with $\chi (p) = - 1$ for the first five primes p, then the corresponding Fekete polynomial $\Sigma _{n = 1}^k\;\chi (n){x^n}$ changes sign on (0, 1). In this paper it is shown that the condition that $\chi (p)$ be -1 for the first four primes p is not sufficient to guarantee such a sign change. More specifically, if $\chi$ is the real nonprincipal character modulo either 1277 or 1973, it is shown that the corresponding Fekete polynomial is positive throughout (0, 1) even though $\chi (2) = \chi (3) = \chi (5) = \chi (7) = - 1$.

Some numerical results on Fekete polynomials

It is known that if χ \chi is a real residue character modulo k with χ ( p ) = − 1 \chi (p) = - 1 for the first five primes p, then the corresponding Fekete polynomial Σ n = 1 k χ ( n ) x n \Sigma _{n = 1}^k\;\chi (n){x^n} changes sign on (0, 1). In this paper it is shown that the condition that χ ( p ) \chi (p) be -1 for the first four primes p is not sufficient to guarantee such a sign change. More specifically, if χ \chi is the real nonprincipal character modulo either 1277 or 1973, it is shown that the corresponding Fekete polynomial is positive throughout (0, 1) even though χ ( 2 ) = χ ( 3 ) = χ ( 5 ) = χ ( 7 ) = − 1 \chi (2) = \chi (3) = \chi (5) = \chi (7) = - 1 .