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The paper deals with the class of entire functions that increase not faster than exp{γ∣z∣6/5(ln∣z∣)−1} and that, together with their first derivatives, take values from a fixed field of algebraic numbers at the points of a two-dimensional lattice of general form (in this case, the values increase not too fast). It is shown that any such functions is either a polynomial or can be represented in the form e−mαzP(eαz), where m is a nonnegative integer, P is a polynomial, and α is an algebraic number. [ABSTRACT FROM AUTHOR]

In the present paper, a polynomial algorithm is suggested for reducing the problem of taking the discrete logarithm in the ring of algebraic integers modulo a power of a prime ideal to a similar problem with the power equal to one. Explicit formulas are obtained; instead of the Fermat quotients, in the case of residues in the ring of rational integers, these formulas use other polynomially computable logarithmic functions, like the $$\mathfrak{p}$$ -adic logarithm. [ABSTRACT FROM AUTHOR]

In the present paper, we consider products of lengths of algebraic numbers whose sum or product is a chosen algebraic number. These products are used to construct a new height function for algebraic numbers. With the help of this function, a metric on the set of all algebraic numbers, which induces the discrete topology, is introduced. [ABSTRACT FROM AUTHOR]

We obtain the system of recurrence relations for rational approximations of the Euler constant generalizing the recurrence relations obtained earlier by Aptekarev with coauthors. The leading coefficient of the recurrence relations of this system is 1, which can be used to verify that the generated numbers are integers. [ABSTRACT FROM AUTHOR]

Let Γ be a directed regular locally finite graph, and let $$\bar \Gamma $$ be the undirected graph obtained by forgetting the orientation of Γ. Let x be a vertex of Γ and let n be a nonnegative integer. We study the length of the shortest directed path in Γ starting at x and ending outside of the ball of radius n centered at x in $$\bar \Gamma $$ . [ABSTRACT FROM AUTHOR]

A multidimensional generalization of the class of Bernstein functions is introduced and the properties of functions belonging to this class are studied. In particular, a new proof of the integral representation of Bernstein functions of several variables is given. Examples are considered. [ABSTRACT FROM AUTHOR]

parametrizing function Sm is introduced. The parametrizing function is a recursive function depending on lexicographic variables, natural variables, and variables whose values are finite sequences of natural variables. Using the function Sm, we construct formulas that provide all the solutions of the equation in a free monoid 〈 a, a, ..., a〉 and only them. [ABSTRACT FROM AUTHOR]

A special matrix semigroup acting on the set of primitive pairs of positive integers naturally parameterizing certainDiophantine equations (such as those defining Pythagorean triples) is constructed and studied. The Pythagorean triples are related to a trichotomy tree of solutions. [ABSTRACT FROM AUTHOR]

The article clarifies and describes the algebraic structure of H• (n; C). The structure is defined as the structure of a supercommulative superalgebra in the space, which is generated by the elements of the Weil group. It states that if the elements of the basis algebra H• (n; C are described in terms of the Weil group it will result into a complicated description in terms of generators and relations may be simply unnecessary for computational purposes .

A Voronin-type discrete universality theorem for the L-functions of new forms is proved. [ABSTRACT FROM AUTHOR]