Related papers: Mahler's problem and Turyn polynomials

This paper addresses a long standing open problem due to Lehmer in which the triple 2,3,7 plays a notable role. Lehmer's problem asks whether there is a gap between 1 and the next smallest algebraic integer with respect to Mahler measure.…

In 1939 P\'al Tur\'an and J\'anos Er\H{o}d initiated the study of lower estimations of maximum norms of derivatives of polynomials, in terms of the maximum norms of the polynomials themselves, on convex domains of the complex plane. As a…

We provide a simple abstract formalism of integration by parts under which we obtain some regularization lemmas. These lemmas apply to any sequence of random variables $(F_n)$ which are smooth and non-degenerated in some sense and enable…

We use the Wilf-Zeilberger method to prove identities between Mahler measures of polynomials. In particular, we offer a new proof of a formula due to Lal\'{i}n, and we show how to translate the identity into a formula involving elliptic…

In a celebrated paper of 1893, Hadamard established the maximal determinant theorem, which establishes an upper bound on the determinant of a matrix with complex entries of norm at most $1$. His paper concludes with the suggestion that…

Let X be a projective integral normal scheme over a number field F; let L be a ample line bundle on X together with a semi-positive adelic metric in the sense of Zhang. The main results of this article are 1) A formula which computes the…

It is well-known that for every $N \geq 1$ and $d \geq 1$ there exist point sets $x_1, \dots, x_N \in [0,1]^d$ whose discrepancy with respect to the Lebesgue measure is of order at most $(\log N)^{d-1} N^{-1}$. In a more general setting,…

We construct a sequence of cyclotomic integers (Gaussian periods) of particularly small Mahler measure/height. We study the asymptotics of their Mahler measure as a function of their conductor, to find that the growth rate is the…

We consider the set $\mathcal{M}_n(\mathbb Z; H)$ of $n\times n$-matrices with integer elements of size at most $H$ and obtain a new upper bound on the number of matrices from $\mathcal{M}_n(\mathbb Z; H)$ with a given characteristic…

An important unsolved question in number theory is the Lehmer's totient problem that asks whether there exists any composite number $n$ such that $\varphi(n)\mid n-1$, where $\varphi$ is the Euler's totient function. It is known that if any…

Mahler equations relate evaluations of the same function $f$ at iterated $b$th powers of the variable. They arise in particular in the study of automatic sequences and in the complexity analysis of divide-and-conquer algorithms. Recently,…

The classical Remez inequality bounds the maximum of the absolute value of a polynomial $P(x)$ of degree $d$ on $[-1,1]$ through the maximum of its absolute value on any subset $Z$ of positive measure in $[-1,1]$. Similarly, in several…

We give an upper bound in O(d ^((n+1)/2)) for the number of critical points of a normal random polynomial with degree d and at most n variables. Using the large deviation principle for the spectral value of large random matrices we obtain…

The excess in electron recoil events reported recently by the XENON1T experiment may be interpreted as evidence for a sizable transition magnetic moment $\mu_{\nu_e\nu_\mu}$ of Majorana neutrinos. We show the consistency of this scenario…

We propose a new algorithm for solving a system of two nonlinear transcendental equations with two complex variables based on the Muller algorithm. The two-dimensional Muller algorithm is tested on systems of different type and is found to…

We consider a version of height on polynomial spaces defined by the integral over the normalized area measure on the unit disk. This natural analog of Mahler's measure arises in connection with extremal problems for Bergman spaces. It…

First we consider the following problem which dates back to Chebyshev, Zolotarev and Achieser: among all trigonometric polynomials with given leading coefficients $a_0,...,a_l,$ $b_0,...,b_l \in \mathbb R$ find that one with least maximum…

Landry, Minsky and Taylor [LMT] introduced two polynomial invariants of veering triangulations -- the taut polynomial and the veering polynomial. We give algorithms to compute these invariants. In their definition [LMT] use only the upper…

Graham and Lov\'{a}sz conjectured in 1978 that the sequence of normalized coefficients of the distance characteristic polynomial of a tree of order $n$ is unimodal with the maximum value occurring at $\lfloor\frac{n}{2}\rfloor$. In this…

The degree-diameter problem seeks to find the largest possible number of vertices in a graph having given diameter and given maximum degree. There has been much recent interest in the problem for mixed graphs, where we allow both undirected…