Zhang-Zagier heights of perturbed polynomials

Zhang-zagier Heights of Perturbed Polynomials

In a previous article we studied the spectrum of the Zhang Zagier height D The progress we made stood on an algorithm that produced polynomials with a small height In this paper we describe a new algorithm that provides even smaller heights It allows us to nd a limit point less than i e better than the previous one namely After some de nitions we detail the principle of the algorithm the result...

On the spectrum of the Zhang-Zagier height

From recent work of Zhang and of Zagier, we know that their height H(α) is bounded away from 1 for every algebraic number α different from 0, 1, 1/2± √ −3/2. The study of the related spectrum is especially interesting, for it is linked to Lehmer’s problem and to a conjecture of Bogomolov. After recalling some definitions, we show an improvement of the so-called ZhangZagier inequality. To achiev...

Heights of CM points I Gross–Zagier formula

Decomposition of perturbed Chebyshev polynomials

We characterize polynomial decomposition fn = r ◦ q with r, q ∈ C[x] of perturbed Chebyshev polynomials defined by the recurrence f0(x) = b, f1(x) = x− c, fn+1(x) = (x− d)fn(x)− afn−1(x), n ≥ 1, where a, b, c, d ∈ R and a > 0. These polynomials generalize the Chebyshev polynomials, which are obtained by setting a = 1/4, c = d = 0 and b ∈ {1, 2}. At the core of the method, two algorithms for pol...

The Zagier Polynomials. Part Ii: Arithmetic Properties of Coefficients