Related papers: Orthogonality of the M\"obius function to polynomi…

We examine correlations of the M\"obius function over $\mathbb{F}_q[t]$ with linear or quadratic phases, that is, averages of the form \begin{equation} \label{eq:average} \frac{1}{q^n}\sum_{\text{deg }f<n} \mu(f)\chi(Q(f)) \end{equation}…

Orthogonal polynomials appear naturally in the study of compositions of M\"obius transformations. In this paper, we consider several classes of orthogonal polynomials associated to non-autonomous perturbations of a parabolic M\"obius map.…

Any function F : {0,...,N-1} -> {-1,1} such that F(x) can be computed from the binary digits of x using a bounded depth circuit is orthogonal to the Mobius function mu in the sense that E_{0 <= x <= N-1} mu(x)F(x) = o(1). The proof combines…

For a fixed polynomial $\Delta$, we study the number of polynomials $f$ of degree $n$ over $\mathbb F_q$ such that $f$ and $f+\Delta$ are both irreducible, an $\mathbb F_q[T]$-analogue of the twin primes problem. In the large-$q$ limit, we…

Let $(X,\nu,T)$ be a measure-preserving system, and let $P_1,\ldots, P_k$ be polynomials with integer coefficients. We prove that, for any $f_1,\ldots, f_k\in L^{\infty}(X)$, the M\"obius-weighted polynomial multiple ergodic averages…

An umbral calculus over local fields of positive characteristic is developed on the basis of a relation of binomial type satisfied by the Carlitz polynomials. Orthonormal bases in the space of continuous $\mathbb F_q$-linear functions are…

Let $F_1,\ldots,F_R$ be homogeneous polynomials of degree $d\ge 2$ with integer coefficients in $n$ variables, and let $\mathbf{F}=(F_1,\ldots,F_R)$. Suppose that $F_1,\ldots,F_R$ is a non-singular system and $n\ge 4^{d+2}d^2R^5$. We prove…

Let $X$ and $Y$ be Banach spaces, let $\mathcal{A}(X)$ stands for the algebra of approximable operators on $X$, and let $P\colon\mathcal{A}(X)\to Y$ be an orthogonally additive, continuous $n$-homogeneous polynomial. If $X^*$ has the…

We consider the enumeration of tropical curves in M\"obius strips for two different lattice structures and relate them to the enumeration of curves in two rational ruled surfaces over a complex elliptic curve. Using this correspondence, we…

Let $c(x_1,...,x_d)$ be a multihomogeneous central polynomial for the $n\times n$ matrix algebra $M_n(K)$ over an infinite field $K$ of positive characteristic $p$. We show that there exists a multihomogeneous polynomial $c_0(x_1,...,x_d)$…

In this chapter are given necessary and sufficient conditions for the regularity of solutions of the functional equation appearing in the theory of classical orthogonal polynomials. In addition, we also present the functional Rodrigues…

We introduce the concept of a bounded below set in a lattice. This can be used to give a generalization of Rota's broken circuit theorem to any finite lattice. We then show how this result can be used to compute and combinatorially explain…

It is known that all $k$-homogeneous orthogonally additive polynomials $P$ over $C(K)$ are of the form $$ P(x)=\int_K x^k d\mu . $$ Thus $x\mapsto x^k$ factors all orthogonally additive polynomials through some linear form $\mu$. We show…

Let $\{\mathbb{P}_n\}_{n\ge 0}$ and $\{\mathbb{Q}_n\}_{n\ge 0}$ be two monic polynomial systems in several variables satisfying the linear structure relation $$\mathbb{Q}_n = \mathbb{P}_n + M_n \mathbb{P}_{n-1}, \quad n\ge 1,$$ where $M_n$…

We show that the sum function of the M\"{o}bius function of a Beurling number system must satisfy the asymptotic bound $M(x)=o(x)$ if it satisfies the prime number theorem and its prime distribution function arises from a monotone…

Let X be a finite set. This paper describes some topological and combinatorial properties of the poset \Omega_X of order relations on X. In particular, the homotopy type of all the intervals in \Omega_X is precisely determined, and the…

We study the growth rate of the summatory function of the M\"obius function in the context of an algebraic curve over a finite field. Our work shows a strong resemblance to its number field counterpart, which was proved by Ng in 2004. We…

The two-matrix model can be solved by introducing bi-orthogonal polynomials. In the case the potentials in the measure are polynomials, finite sequences of bi-orthogonal polynomials (called "windows") satisfy polynomial ODEs as well as…

We prove locally uniform spacing for the zeros of orthogonal polynomials on the real line under weak conditions (Jacobi parameters approach the free ones and are of bounded variation). We prove that for ergodic discrete Schrodinger…

Orthogonal polynomials with respect to a weight function defined on a wedge in the plane are studied. A basis of orthogonal polynomials is explicitly constructed for two large class of weight functions and the convergence of Fourier…