Cubic symmetric polynomials yielding variations of the Erdős–Ginzburg–Ziv theorem

Let p be a prime and let $\varphi\in\mathbb{Z}_{p}[x_{1},x_{2},\ldots, x_{p}]$ be a symmetric polynomial, where  $\mathbb {Z}_{p}$ is the field of p elements. A sequence T in  $\mathbb {Z}_{p}$ of length p is called a φ-zero sequence if φ(T)=0; a sequence in $\mathbb {Z}_{p}$ is called a φ-zero free sequence if it does not contain any φ-zero subsequence. Motivated by the EGZ theorem for the prime p, we consider symmetric polynomials $\varphi\in \mathbb {Z}_{p}[x_{1},x_{2},\ldots, x_{p}]$ , which satisfy the following two conditions: (i) every sequence in  $\mathbb {Z}_{p}$ of length 2p?1 contains a φ-zero subsequence, and (ii) the φ-zero free sequences in  $\mathbb {Z}_{p}$ of maximal length are all those containing exactly two distinct elements, where each element appears p?1 times. In this paper, we determine all symmetric polynomials in $\mathbb {Z}_{p}[x_{1},x_{2},\ldots, x_{p}]$ of degree not exceeding 3 satisfying the conditions above.     

Heavy tail phenomenon and convergence to stable laws for iterated Lipschitz maps

We consider the Markov chain ${\{X_n^x\}_{n=0}^\infty}$ on ${\mathbb{R}^d}$ defined by the stochastic recursion ${X_{n}^{x}= \psi_{\theta_{n}} (X_{n-1}^{x})}$ , starting at ${x\in\mathbb{R}^d}$ , where ?? ₁, ?? ₂, . . . are i.i.d. random variables taking their values in a metric space ${(\Theta, \mathfrak{r})}$ , and ${\psi_{\theta_{n}} :\mathbb{R}^d\mapsto\mathbb{R}^d}$ are Lipschitz maps. Assume that the Markov chain has a unique stationary measure ??. Under appropriate assumptions on ${\psi_{\theta_n}}$ , we will show that the measure ?? has a heavy tail with the exponent ???>?0 i.e. ${\nu(\{x\in\mathbb{R}^d: |x| > t\})\asymp t^{-\alpha}}$ . Using this result we show that properly normalized Birkhoff sums ${S_n^x=\sum_{k=1}^n X_k^x}$ , converge in law to an ??-stable law for ${\alpha\in(0, 2]}$ .     

Value-sets of polynomials at p-adic integers

In this paper we are concerned with the classification of the subsets A of ${\mathbb{Z}_p}$ which occur as images ${f(\mathbb{Z}_p^r)}$ of polynomial functions ${f:\mathbb{Z}_p^r\to \mathbb{Z}_p}$ , limiting ourselves to compact-open subsets (i.e. finite unions of open balls). We shall prove three main results: (i) Every compact-open ${A\subset \mathbb{Z}_p}$ is of the shape ${A=f(\mathbb{Z}_p^r)}$ for suitable r and ${f\in\mathbb{Z}_p[X_1,\ldots ,X_r]}$ . (ii) For each r ₀ there is a compact-open A such that in (i) we cannot take r < r ₀. (iii) For any compact-open set ${A\subset \mathbb{Z}_p}$ there exists a polynomial ${f\in\mathbb{Q}_p[X]}$ such that ${f(\mathbb{Z}_p)=A}$ . We shall also discuss in more detail which sets A can be represented as ${f(\mathbb{Z}_p)}$ for a polynomial ${f\in\mathbb{Z}_p[X]}$ in a single variable.     

Localization in One-dimensional Quasi-periodic Nonlinear Systems

To investigate localization in one-dimensional quasi-periodic nonlinear systems, we consider the Schrödinger equation $${\rm i}\dot{q}_n+\epsilon(q_{n+1}+q_{n-1})+V(n\tilde{\alpha}+x)q_n+ |q_n|^2q_n=0,\quad n\in\mathbb{Z},$$ as a model, with V a nonconstant real-analytic function on ${\mathbb{R}/\mathbb{Z}}$ , and ${\tilde{\alpha}}$ satisfying a certain Diophantine condition. It is shown that, if ${\epsilon}$ is sufficiently small, then for a.e. ${x\in\mathbb{R}/\mathbb{Z}}$ , dynamical localization is maintained for “typical” solutions in a quasi-periodic time-dependent way.     

Pfaffian representations of cubic surfaces

Let $\mathbb{K }$ be a field of characteristic zero. We describe an algorithm which requires a homogeneous polynomial $F$ of degree three in $\mathbb{K }[x_{0},x_1,x_{2},x_{3}]$ and a zero ${\mathbf{a }}$ of $F$ in $\mathbb{P }^{3}_{\mathbb{K }}$ and ensures a linear Pfaffian representation of $\text{ V}(F)$ with entries in $\mathbb{K }[x_{0},x_{1},x_{2},x_{3}]$ , under mild assumptions on $F$ and ${\mathbf{a }}$ . We use this result to give an explicit construction of (and to prove the existence of) a linear Pfaffian representation of $\text{ V}(F)$ , with entries in $\mathbb{K }^{\prime }[x_{0},x_{1},x_{2},x_{3}]$ , being $\mathbb{K }^{\prime }$ an algebraic extension of $\mathbb{K }$ of degree at most six. An explicit example of such a construction is given.     

Uniform existential interpretation of arithmetic in rings of functions of positive characteristic

We show that first order integer arithmetic is uniformly positive-existentially interpretable in large classes of (subrings of) function fields of positive characteristic over some languages that contain the language of rings. One of the main intermediate results is a positive existential definition (in these classes), uniform among all characteristics p, of the binary relation “ $y=x^{p^{s}}$ or $x=y^{p^{s}}$ for some integer s≥0”. A natural consequence of our work is that there is no algorithm to decide whether or not a system of polynomial equations over $\mathbb {Z}[z]$ has solutions in all but finitely many polynomial rings $\mathbb {F}_{p}[z]$ . Analogous consequences are deduced for the rational function fields $\mathbb {F}_{p}(z)$ , over languages with a predicate for the valuation ring at zero.     

Square-free values of f(p), f cubic

Let \({f \in \mathbb{Z}[x]}\) , \({\deg f =3}\) . Assume that f does not have repeated roots. Assume as well that, for every prime q, \({f(x)\not\equiv 0}\) mod q ² has at least one solution in \({(\mathbb{Z}/q^2 \mathbb{Z})^*}\) . Then, under these two necessary conditions, there are infinitely many primes p such that f(p) is square-free.     

The Essential Norm of a Generalized Composition Operator Between Bloch-Type Spaces and Q K Type Spaces

Let ?? be an analytic self-map of the unit disk ${\rm \mathbb{D},H(\rm \mathbb{D})}$ the space of analytic functions on ${{\rm \mathbb{D}}}$ and ${g \in H(\rm \mathbb{D})}$ . We define a linear operator as follows $$C_\varphi^gf(z)=\int\limits_0^zf'(\varphi(w))g(w)\, {\rm d}w, $$ on ${ H(\rm \mathbb{D})}$ . In this paper, estimates for the essential norm of the generalized composition operator between Bloch-type spaces and Q _K type spaces are obtained.     

Minimizing and maximizing the Euclidean norm of the product of two polynomials

We consider the problem of minimizing or maximizing the quotient $$f_{m,n}(p,q):=\frac{\|{pq}\|}{\|{p}\|\|{q}\|} \ ,$$ where $p=p_0+p_1x+\dots+p_mx^m$ , $q=q_0+q_1x+\dots+q_nx^n\in{\mathbb K}[x]$ , ${\mathbb K}\in\{{\mathbb R},{\mathbb C}\}$ , are non-zero real or complex polynomials of maximum degree $m,n\in{\mathbb N}$ respectively and $\|{p}\|:=(|p_0|^2+\dots+|p_m|^2)^{\frac{1}{2}}$ is simply the Euclidean norm of the polynomial coefficients. Clearly f _m,n is bounded and assumes its maximum and minimum values min f _m,n ?=?f _m,n (p _min, q _min) and max f _m,n ?=?f(p _max, q _max). We prove that minimizers p _min, q _min for ${\mathbb K}={\mathbb C}$ and maximizers p _max, q _max for arbitrary ${\mathbb K}$ fulfill $\deg(p_{\min})=m=\deg(p_{\max})$ , $\deg(q_{\min})=n=\deg(q_{\max})$ and all roots of p _min, q _min, p _max, q _max have modulus one and are simple. For ${\mathbb K}={\mathbb R}$ we can only prove the existence of minimizers p _min, q _min of full degree m and n respectively having roots of modulus one. These results are obtained by transferring the optimization problem to that of determining extremal eigenvalues and corresponding eigenvectors of autocorrelation Toeplitz matrices. By the way we give lower bounds for min f _m,n for real polynomials which are slightly better than the known ones and inclusions for max f _m,n .     

A Congruence Connecting Latin Rectangles and Partial Orthomorphisms

A partial orthomorphism of ${\mathbb{Z}_{n}}$ is an injective map ${\sigma : S \rightarrow \mathbb{Z}_{n}}$ such that ${S \subseteq \mathbb{Z}_{n}}$ and ??(i)?Ci ? ??(j)? j (mod n) for distinct ${i, j \in S}$ . We say ?? has deficit d if ${|S| = n - d}$ . Let ??(n, d) be the number of partial orthomorphisms of ${\mathbb{Z}_{n}}$ of deficit d. Let ??(n, d) be the number of partial orthomorphisms ?? of ${\mathbb{Z}_n}$ of deficit d such that ??(i) ? {0, i} for all ${i \in S}$ . Then ??(n, d) =???(n, d)n ²/d ² when ${1\,\leqslant\,d < n}$ . Let R _{k, n} be the number of reduced k ×?n Latin rectangles. We show that $$R_{k, n} \equiv \chi (p, n - p)\frac{(n - p)!(n - p - 1)!^{2}}{(n - k)!}R_{k-p,\,n-p}\,\,\,\,(\rm {mod}\,p)$$ when p is a prime and ${n\,\geqslant\,k\,\geqslant\,p + 1}$ . In particular, this enables us to calculate some previously unknown congruences for R _{n, n} . We also develop techniques for computing ??(n, d) exactly. We show that for each a there exists??? _a such that, on each congruence class modulo??? _a , ??(n, n-a) is determined by a polynomial of degree 2a in n. We give these polynomials for ${1\,\leqslant\,a\,\leqslant 6}$ , and find an asymptotic formula for ??(n, n-a) as n ?? ??, for arbitrary fixed a.     

Sturmian maximizing measures for the piecewise-linear cosine family

Let T be the angle-doubling map on the circle $\mathbb{T}$ , and consider the 1-parameter family of piecewise-linear cosine functions $f_\theta :\mathbb{T} \to \mathbb{R}$ , defined by $f_\theta (x) = 1 - 4d_\mathbb{T} (x,\theta )$ . We identify the maximizing T-invariant measures for this family: for each ?? the f _?? -maximizing measure is unique and Sturmian (i.e. with support contained in some closed semi-circle). For rational p/q, we give an explicit formula for the set of functions in the family whose maximizing measure is the Sturmian measure of rotation number p/q. This allows us to analyse the variation with ?? of the maximum ergodic average for f _?? .     

On Simultaneous Digital Expansions of Polynomial Values

Let s _q denote the q-ary sum-of-digits function and let \({P_1(X), P_2(X) \in \mathbb{Z}[X]}\) with \({P_1(\mathbb{N}), P_2(\mathbb{N}) \subset \mathbb{N}}\) be polynomials of degree \({h, l \geqq 1, h \neq l}\) , respectively. In this note we show that ( \({s_q(P_1(n))/s_q(P_2(n)))_{n \geqq 1}}\) is dense in \({\mathbb{R}^+}\) . This extends work by Stolarsky [9] and Hare, Laishram and Stoll [6].     

A construction of integer-valued polynomials with prescribed sets of lengths of factorizations

For an arbitrary finite non-empty set $S$ of natural numbers greater $1$ , we construct $f\in \text{ Int }(\mathbb{Z })=\{g\in \mathbb{Q }[x]\mid g(\mathbb{Z })\subseteq \mathbb{Z }\}$ such that $S$ is the set of lengths of $f$ , i.e., the set of all $n$ such that $f$ has a factorization as a product of $n$ irreducibles in $\text{ Int }(\mathbb{Z })$ . More generally, we can realize any finite non-empty multi-set of natural numbers greater 1 as the multi-set of lengths of the essentially different factorizations of $f$ .     

Postulation of Disjoint Unions of Lines and Double Lines in {\mathbb{P}^3}

A double line ${C \subset \mathbb{P}^3}$ is a connected divisor of type (2, 0) on a smooth quadric surface. Fix ${(a, c) \in \mathbb{N}^2\ \backslash\ \{(0, 0)\}}$ . Let ${X \subset \mathbb{P}^3}$ be a general disjoint union of a lines and c double lines. Then X has maximal rank, i.e. for each ${t \in \mathbb{Z}}$ either ${h^1(\mathcal{I}_X(t)) = 0}$ or ${h^0(\mathcal{I}_X(t)) = 0}$ .     

Poisson kernel and Cauchy formula of a non-symmetric transitive domain

In 1965, Lu Yu-Qian discovered that the Poisson kernel of the homogenous domain S m,p,q={Z∈Cm×m, Z1∈Cm×p,Z2 ∈Cq×m|2i1( Z-Z+)-Z1Z1′-Z2′Z20} does not satisfy the Laplace-Beltrami equation associated with the Bergman metric when S m,p,q is not symmetric. However the map T0:Z→Z, Z1→Z1 , Z2→Z2 transforms S m,p,q into a domain S I (m, m + p + q) which can be mapped by the Cayley transformation into the classical domains R I (m, m + p + q). The pull back of the Bergman metric of R I (m, m + p + q) to S m,p,q is a Riemann metric ds 2 which is not a Khler metric and even not a Hermitian metric in general. It is proved that the Laplace-Beltrami operator associated with the metric ds 2 when it acts on the Poisson kernel of S m,p,q equals 0. Consequently, the Cauchy formula of S m,p,q can be obtained from the Poisson formula.     

Central limit theorems for moving average processes*

Let $ {{\left( {{\xi_n}} \right)}_{{n\in \mathbb{Z}}}} $ be a stationary sequence of real random variables with E ξ ₀ = 0 and infinite variance. Furthermore, assume that $ {{\left( {{c_n}} \right)}_{{n\in \mathbb{Z}}}} $ is a sequence of real numbers and $ {X_n}=\sum {_{{j\in \mathbb{Z}}}{c_j}{\xi_{n-j }}} $ is a moving average processes driven by $ {{\left( {{\xi_n}} \right)}_{{n\in \mathbb{Z}}}} $ . By using a decomposition of the moving average processes, a central limit theorem for the partial sums $ \sum\nolimits_{k=1}^n {{X_k}} $ is established. As applications, we obtain some central limit theorems for stationary dependent sequences $ {{\left( {{\xi_n}} \right)}_{{n\in \mathbb{Z}}}} $ , such as associated sequence, martingale difference, and so on.     

Singular limits solution for two-dimensional elliptic problems involving exponential nonlinearities with nonlinear gradient terms and singular weights

Given Ω bounded open regular set of ${\mathbb{R}^2}$ , ${q_1,\ldots, q_K \in \Omega}$ , ${\varrho : \Omega \longrightarrow [0,+\infty)}$ a regular bounded function and ${V: \Omega \longrightarrow [0,+\infty)}$ a bounded fuction. We give a sufficient condition for the model problem $$(P):\qquad-{\Delta}u -{\lambda}\varrho(x)|{\nabla}u|^2 = \varepsilon^{2}V(x)e^u$$ to have a positive weak solution in Ω with u = 0 on ?Ω, which is singular at each q _i as the parameters ${\varepsilon}$ and λ tend to 0, essentially when the set of concentration points q _i and the set of zeros of V are not necessarily disjoint.     

Quadratic polynomials represented by norm forms

Let ${P(t) \in \mathbb{Q}[t]}$ be an irreducible quadratic polynomial and suppose that K is a quartic extension of ${\mathbb{Q}}$ containing the roots of P(t). Let ${{\bf N}_{K/\mathbb{Q}}({\rm x})}$ be a full norm form for the extension ${K/\mathbb{Q}}$ . We show that the variety $$\begin{array}{ll}P(t)={\bf N}_{K/\mathbb{Q}}({\rm x})\neq 0\end{array}$$ satisfies the Hasse principle and weak approximation. The proof uses analytic methods.     

Divisibility of class numbers of imaginary quadratic function fields by a fixed odd number

In this paper we find a new lower bound on the number of imaginary quadratic extensions of the function field $\mathbb{F}_{q}(x)$ whose class groups have elements of a fixed odd order. More precisely, for q, a power of an odd prime, and g a fixed odd positive integer ≥?3, we show that for every ε?>?0, there are $\gg q^{L(\frac{1}{2}+\frac{3}{2(g+1)}-\epsilon)}$ polynomials $f \in \mathbb{F}_{q}[x]$ with $\deg f=L$ , for which the class group of the quadratic extension $\mathbb{F}_{q}(x, \sqrt{f})$ has an element of order g. This sharpens the previous lower bound $q^{L(\frac{1}{2}+\frac{1}{g})}$ of Ram Murty. Our result is a function field analogue which is similar to a result of Soundararajan for number fields.