Small zeros of quadratic forms with linear conditions

Given a quadratic form and M linear forms in N+1 variables with coefficients in a number field K, suppose that there exists a point in K^N+1 at which the quadratic form vanishes and all the linear forms do not. Then we show that there exists a point like this of relatively small height. This generalizes a result of D.W. Masser.     

Diophantine equations of Pellian type

We investigate the solutions of diophantine equations of the form dx²−d^?y²=±t for t∈{1,2,4} and their connections with ideal theory, continued fractions and Jacobi symbols.     

Parametrization of Pythagorean triples by a single triple of polynomials

It is well known that Pythagorean triples can be parametrized by two triples of polynomials with integer coefficients. We show that no single triple of polynomials with integer coefficients in any number of variables is sufficient, but that there exists a parametrization of Pythagorean triples by a single triple of integer-valued polynomials.     

Algebraic independence of arithmetic gamma values and Carlitz zeta values

We consider the values at proper fractions of the arithmetic gamma function and the values at positive integers of the zeta function for Fq[θ] and provide complete algebraic independence results for them.     

On the quadratic character of quadratic units

Let be a prime. Let a,b∈Z with p?a(a²+b²). In the paper we mainly determine by assuming p=c²+d² or p=Ax²+2Bxy+Cy² with AC−B²=a²+b². As an application we obtain simple criteria for εD to be a quadratic residue , where D>1 is a squarefree integer such that D is a quadratic residue of p, εD is the fundamental unit of the quadratic field with negative norm. We also establish the congruences for and obtain a general criterion for p|U(p−1)/4, where {Un} is the Lucas sequence defined by U₀=0, U₁=1 and Un+1=bUn+k²Un−1(n?1).     

Determination of algebraic relations among special zeta values in positive characteristic

As an analogue to special values at positive integers of the Riemann zeta function, we consider Carlitz zeta values ζC(n) at positive integers n. By constructing t-motives after Papanikolas, we prove that the only algebraic relations among these characteristic p zeta values are those coming from the Euler-Carlitz relations and the Frobenius pth power relations.     

On consecutive happy numbers

Let e?1 and b?2 be integers. For a positive integer with 0?aj<b, define

     

Complete solution of the polynomial version of a problem of Diophantus

In this paper, we prove that if {a,b,c,d} is a set of four non-zero polynomials with integer coefficients, not all constant, such that the product of any two of its distinct elements plus 1 is a square of a polynomial with integer coefficients, then

(a+b−c−d)2=4(ab+1)(cd+1).     

Newman's phenomenon for generalized Thue-Morse sequences

Let t_j=(-1)^s(j) be the Thue-Morse sequence with s(j) denoting the sum of the digits in the binary expansion of j. A well-known result of Newman [On the number of binary digits in a multiple of three, Proc. Amer. Math. Soc. 21 (1969) 719-721] says that t₀+t₃+t₆+?+t_3k>0 for all k?0.In the first part of the paper we show that t₁+t₄+t₇+?+t_3k+1<0 and t₂+t₅+t₈+?+t_3k+2?0 for k?0, where equality is characterized by means of an automaton. This sharpens results given by Dumont [Discrépance des progressions arithmétiques dans la suite de Morse, C. R. Acad. Sci. Paris Sér. I Math. 297 (1983) 145-148]. In the second part we study more general settings. For a,g?2 let ω_a=exp(2πi/a) and , where s_g(j) denotes the sum of digits in the g-ary digit expansion of j. We observe trivial Newman-like phenomena whenever a|(g-1). Furthermore, we show that the case a=2 inherits many Newman-like phenomena for every even g?2 and large classes of arithmetic progressions of indices. This, in particular, extends results by Drmota and Ska?ba [Rarified sums of the Thue-Morse sequence, Trans. Amer. Math. Soc. 352 (2000) 609-642] to the general g-case.     

Metric Diophantine approximation over a local field of positive characteristic

The conjectures of Sprind?uk in the metric theory of Diophantine approximation are established over a local field of positive characteristic. In the real case, these were settled by D. Kleinbock and G.A. Margulis using a new technique which involved nondivergence estimates for quasi-polynomial flows on the space of lattices. We extend their technique to the positive characteristic setting.     

Continued fractions, special values of the double sine function, and Stark units over real quadratic fields

Let F be a real quadratic field and m an integral ideal of F. Two Stark units, εm,1 and εm,2, are conjectured to exist corresponding to the two different embeddings of F into R. We define new ray class invariants and associated to each class C₊ of the narrow ray class group modulo m and dependent separately on the two different embeddings of F into R. These invariants are defined as a product of special values of the double sine function in a compact and canonical form using a continued fraction approach due to Zagier and Hayes. We prove that both Stark units εm,1 and εm,2, assuming they exist, can be expressed simultaneously and symmetrically in terms of and , thus giving a canonical expression for every existent Stark unit over F as a product of double sine function values. We prove that Stark units do exist as predicted in certain special cases.     

Cubic residues and binary quadratic forms

Let p>3 be a prime, u,v,d∈Z, gcd(u,v)=1, p?u²−dv² and , where is the Legendre symbol. In the paper we mainly determine the value of by expressing p in terms of appropriate binary quadratic forms. As applications, for we obtain a general criterion for and a criterion for εd to be a cubic residue of p, where εd is the fundamental unit of the quadratic field . We also give a general criterion for , where {Un} is the Lucas sequence defined by U₀=0, U₁=1 and Un+1=PUn−QUn−1 (n?1). Furthermore, we establish a general result to illustrate the connections between cubic congruences and binary quadratic forms.     

On character sums of binary quadratic forms

We establish character sum bounds of the form

     

Thue equations and torsion groups of elliptic curves

A new characterization of rational torsion subgroups of elliptic curves is found, for points of order greater than 4, through the existence of solution for systems of Thue equations.     

The zeros of a quadratic form at square-free points

Let F(x₁,…,xn) be a nonsingular indefinite quadratic form, n=3 or 4. For n=4, suppose the determinant of F is a square. Results are obtained on the number of solutions of

F(x₁,…,xn)=0     

On the Diophantine equation ax + by = ck

In this paper we investigate the solvability and the representation of the solutions of the equation ax² +by² = ckⁿ. We extend and improve many known results. In particular, we completely solve the equation (a ± 1)x² + (3a ? 1) = 4aⁿ, 2 ? n.     

Existence of solutions of non-autonomous second order functional differential equations with infinite delay

In this paper the existence of solutions of a non-autonomous abstract retarded functional differential equation of second order with infinite delay is considered. Assuming the existence of an evolution operator corresponding to the associate abstract Cauchy problem of second order, we establish the existence of mild solutions of the functional equation. Furthermore, we study the existence of classical solutions of the abstract Cauchy problem of second order and we apply these results to establish the existence of classical solutions of the functional equation. Finally, we apply our results to study the existence of solutions of the non-autonomous wave equation with delay.     

Complete decomposition of Dickson-type polynomials and related Diophantine equations

We characterize decomposition over C of polynomials defined by the generalized Dickson-type recursive relation (n?1)