Some Pólya-Type Irreducibility Criteria for Multivariate Polynomials

We provide irreducibility criteria for multivariate polynomials with coefficients in an arbitrary field that extend a classical result of Pólya for polynomials with integer coefficients. In particular, we provide irreducibility conditions for polynomials of the form f(X)(Y ? f ₁(X))…(Y ? f _n (X)) + g(X), with f, f ₁, ?, f _n , g univariate polynomials over an arbitrary field.     

On the Computation of Powers of Sparse Polynomials

This paper examines the time for computing integer powers of a class of polynomials in one or several variables. A member f of the class of “sparse” polynomials can be characterized by the number of nonzero terms, t, in the representation of f. Where, for power 4, the most obvious methods result in multiplications, a method is presented which requires fewer than multiplications. This algorithm requires, for t » n » 1 about tⁿ/n! + O(t^{n ? 1} /(2(n ? 2)!)) multiplications, which is proved to be a lower-bound on the computation.     

Period Integrals and Rankin–Selberg L-functions on GL(n)

We compute the second moment of a certain family of Rankin–Selberg L-functions L(f ×?g, 1/2) where f and g are Hecke–Maass cusp forms on GL(n). Our bound is as strong as the Lindel?f hypothesis on average, and recovers individually the convexity bound. This result is new even in the classical case n?=?2.     

Generalization of a theorem of Erdős and Rényi on Sidon sequences

Erd?s and Rényi claimed and Vu proved that for all h ≥ 2 and for all ? > 0, there exists g = g_h(?) and a sequence of integers A such that the number of ordered representations of any number as a sum of h elements of A is bounded by g, and such that |A ∩ [1,x]| ? x^1/h‐?. We give two new proofs of this result. The first one consists of an explicit construction of such a sequence. The second one is probabilistic and shows the existence of such a g that satisfies g_h(?) ? ?^?1, improving the bound g_h(?) ? ?^?h+1 obtained by Vu. Finally we use the “alteration method” to get a better bound for g₃(?), obtaining a more precise estimate for the growth of B₃[g] sequences. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 2010     

Enumeration of arrays of a given size

Enumeration of arrays whose row and column sums are specified have been studied by a number of people. It has been determined that the function that enumerates square arrays of dimension n, whose rows and columns sum to a fixed non-negative integer r, is a polynomial in r of degree (n ? 1)².In this paper we consider rectangular arrays whose rows sum to a fixed non-negative integer r and whose columns sum to a fixed non-negative integer s, determined by ns = mr. in particular, we show that the functions which enumerate 2 × n and 3 × n arrays with fixed row sums $\text{nr}\text{(2, n)}$ and $\text{nr}\text{(3, n)}$, where the symbol (a, b) denotes the greatest common divisor of a and b, and fixed column sums, are polynomials in r of degrees (n ? 1) and 2(n ? 1) respectively. We have found simple formulas to evaluate these polynomials for negative values, - r, and we show that for certain small negative integers our polynomials will always be zero. We also considered the generating functions of these polynomials and show that they are rational functions of degrees less than zero, whose denominators are of the forms (1 ? y)ⁿ and (1 ? y)²ⁿ?1 respectively and whose numerators are polynomials in y whose coefficients satisfy certain properties. In the last section we list the actual polynomials and generating functions in the 2 × n and 3 × n cases for small specific values of n.     

On approximation of convex functions by rational ones in integral metrics

Пустьf - действительн означная конечная фу нкция на конечном отрезке Δ=[а, b] вещественной оси, |Δ|=b?a, M(f) = sup {|f(x)|: x∈Δ}, R_n(f,p Δ) = inf∥f?r∥_Lp(Δ) (0 < p < ∞), где нижняя грань бере тся по всем рациональ ным функциямr порядка не вышеп, K(М, Δ) класс всех выпуклых на отре зке Δ функцийf, для кот орыхM(f)≦M. Теорема.При любом вещ ественном р, 0<р<∞ и вс ехп=1, 2, ... sup {R_n(f, p, Δ):f∈K(M, Δ)} ≦ C(p)M|Δ|^1/pn^?2,где С(р) - величина, зави сящая лишь от р.     

The least common multiple of a sequence of products of linear polynomials

Let f(x) be the product of several linear polynomials with integer coefficients. In this paper, we obtain the estimate log?lcm?(f(1),…,f(n))～An as n→∞, where A is a constant depending on?f.     

Zur Abschätzung des Bestapproximationsfehlers bei der Approximation differenzierbarer Funktionen durch Polynome

The problem attacked here is to find smaller factorsγ (n,?),n > ?, for the inequality $$\delta _n [f] \leqq \gamma (n,\varrho ) \cdot \mathop {\sup }\limits_{x \in [ - 1,1]} |f^{(\varrho )} (x)|$$ than are already known. Heref(^? (x) denotes the?-th derivative off(x), andδ _n [f] is the error of the best Chebyshev-approximation off by algebraic polynomials of degree ≦n. A new approach to this problem is demonstrated and the results we got forn≦9,?≦8 by the use of a computer are presented.     

On the irreducibility of a class of polynomials,III

This work is a continuation and extension of our earlier articles on irreducible polynomials. We investigate the irreducibility of polynomials of the form g(f(x)) over an arbitrary but fixed totally real algebraic number field $\text{L}$, where g(x) and f(x) are monic polynomials with integer coefficients in $\text{L}$, g is irreducible over $\text{L}$ and its splitting field is a totally imaginary quadratic extension of a totally real number field. A consequence of our main result is as follows. If g is fixed then, apart from certain exceptions f of bounded degree, g(f(x)) is irreducible over $\text{L}$ for all f having distinct roots in a given totally real number field.     

Comparison of the best uniform approximations of analytic functions in the disk and on its boundary

Denote by C _A the set of functions that are analytic in the disk |z| < 1 and continuous on its closure |z| ≤ 1; let ? _n , n = 0, 1, 2, ..., be the set of rational functions of degree at most n. Denote by R _n (f) (R _n (f) _A ) the best uniform approximation of a function f ∈ C _A on the circle |z| = 1 (in the disk |z| ≤ 1) by the set ? _n . The following equality is proved for any n ≥ 1: sup{R _n (f) _A /R _n (f): f ∈ C _A ? ? _n } = 2. We also consider a similar problem of comparing the best approximations of functions in C _A by polynomials and trigonometric polynomials.     

Binding number and minimum degree for the existence of (g,f,n)-critical graphs

Let G be a graph of order p. The binding number of G is defined as $\mbox{bind}(G):=\min\{\frac{|N_{G}(X)|}{|X|}\mid\emptyset\neq X\subseteq V(G)\,\,\mbox{and}\,\,N_{G}(X)\neq V(G)\}$ . Let g(x) and f(x) be two nonnegative integer-valued functions defined on V(G) with g(x)≤f(x) for any x∈V(G). A graph G is said to be (g,f,n)-critical if G?N has a (g,f)-factor for each N?V(G) with |N|=n. If g(x)≡a and f(x)≡b for all x∈V(G), then a (g,f,n)-critical graph is an (a,b,n)-critical graph. In this paper, several sufficient conditions on binding number and minimum degree for graphs to be (a,b,n)-critical or (g,f,n)-critical are given. Moreover, we show that the results in this paper are best possible in some sense.     

Dirichlet series of Rankin-Cohen brackets

Given modular forms f and g of weights k and ?, respectively, their Rankin-Cohen bracket corresponding to a nonnegative integer n is a modular form of weight k+?+2n, and it is given as a linear combination of the products of the form f(r)g(n−r) for 0?r?n. We use a correspondence between quasimodular forms and sequences of modular forms to express the Dirichlet series of a product of derivatives of modular forms as a linear combination of the Dirichlet series of Rankin-Cohen brackets.     

Равносходимость разложений в кратный ряд и интеграл фурье, «прямоугольные частичные суммы» которых рассматриваются по некоторой подпоследовательности

In this paper we investigate the problem of the equiconvergence on T ^N = [-π, π) ^N of the expansions in multiple trigonometric series and Fourier integral of functions f ∈ L _p (T ^N ) and g ∈ L _p (? ^N ), where p > 1, N ≥ 3, g(x) = f(x) on T ^N , in the case when the “rectangular partial sums” of the indicated expansions, i.e.,– _n (x; f) and J _α(x; g), respectively, have indices n ∈ ? ^N and α ∈ ? ^N (n _j = [α _j ], j = 1,...,N, [t] is the integer part of t ∈ ?¹), in those certain components are the elements of “lacunary sequences”.     

Selection principle for pointwise bounded sequences of functions

For a number ? > 0 and a real function f on an interval [a, b], denote by N(?, f, [a, b]) the least upper bound of the set of indices n for which there is a family of disjoint intervals [a _i , b _i ], i = 1, …, n, on [a, b] such that |f(a _i ) ? f(b _i )| > ? for any i = 1, …, n (sup Ø = 0). The following theorem is proved: if {f _j } is a pointwise bounded sequence of real functions on the interval [a, b] such that n(?) ≡ lim sup _j→∞ N(?, f _j , [a, b]) < ∞ for any ? > 0, then the sequence {f _j } contains a subsequence which converges, everywhere on [a, b], to some function f such that N(?, f, [a, b]) ≤ n(?) for any ? > 0. It is proved that the main condition in this theorem related to the upper limit is necessary for any uniformly convergent sequence {f _j } and is “almost” necessary for any everywhere convergent sequence of measurable functions, and many pointwise selection principles generalizing Helly’s classical theorem are consequences of our theorem. Examples are presented which illustrate the sharpness of the theorem.     

Degrees and cycles in digraphs

In this article, we give conditions on the total degrees of the vertices in a strong digraph implying the existence of a cycle of length at least $\text{?}\text{(n?1)}\text{h}\text{? + 1}$, where n is the number of vertices of the graph and h an integer, 1?h?n?1. The same conditions imply the existence of a path of length $\text{?}\text{(n?1)}\text{h}\text{? + ?}\text{(n?2)}\text{h}\text{?}$. In the case of strong oriented graphs (antisymmetric digraphs) we improve these conditions. In both cases, we show that the given conditions are the best possible.     

On oscillation of solutions of higher order forced functional differential inequalities

Oscillation criteria for the class of forced functional differential inequalities x(t){L_nx(t) + f(t, x(t), x[g₁(t)],…, x[g_m(t)]) ? h(t)} ? 0, for n even, and x(t){L_nx(t) ? f(t, x(t), x[g₁(t)],…, x[g_m(t)]) ? h(t)} ? 0, for n odd, are established.     

Global existence and blowup solutions for quasilinear parabolic equations

The authors discuss the quasilinear parabolic equation ut=∇⋅(g(u)∇u)+h(u,∇u)+f(u) with u_|∂Ω=0, u(x,0)=?(x). If f, g and h are polynomials with proper degrees and proper coefficients, they show that the blowup property only depends on the first eigenvalue of −Δ in Ω with Dirichlet boundary condition. For a special case, they obtain a sharp result.     

On simple modules of Hecke algebras of type Dn

This is a continuation of our previous work. We classify all the simple ?_q(D _n )-modules via an automorphismh defined on the set { λ | D^λ ≠ 0}. Whenf _n(q) ≠ 0, this yields a classification of all the simple ? ^q (D _n)- modules for arbitrary n. In general ( i. e., q arbitrary), if λ⁽¹⁾ = λ⁽²⁾,wegivea necessary and sufficient condition ( in terms of some polynomials ) to ensure that the irreducible ?_q,1(B _n )- module D^λ remains irreducible on restriction to ?_q(D _n ).     

Stability of a simple Levi–Civitá functional equation on non-unital commutative semigroups

In this paper, we study the Hyers–Ulam stability of a simple Levi–Civitá functional equation f(x+y)=f(x)h(y)+f(y) and its pexiderization f(x+y)= g(x) h(y)+k(y) on non-unital commutative semigroups by investigating the functional inequalities |f(x+y)?f(x)h(y)?f(y)|≤?? and |f(x+y)?g(x)h(y)?k(y)|≤??, respectively. We also study the bounded solutions of the simple Levi–Civitá functional inequality.     

Birational composition of quadratic forms over a local field

Let f(X) and g(Y) be nondegenerate quadratic forms of dimensions m and n, respectively, over K, char K ≠ 2. The problem of birational composition of f(X) and g(Y) is considered: When is the product f(X) · g(Y) birationally equivalent over K to a quadratic form h(Z) over K of dimension m + n? The solution of the birational composition problem for anisotropic quadratic forms over K in the case of m = n = 2 is given. The main result of the paper is the complete solution of the birational composition problem for forms f(X) and g(Y) over a local field P, char P ≠ 2.