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The aim of this paper is to characterize the ( β , A )-representations of a real number where the base β is a quadratic integer and its conjugate β ′ satisfy: 1 < | β ′ | ≤ β . We prove that if there exist a representation of a real number x satisfies some streaming conditions, then x either belongs to Q (β) , or is transcendental. [ABSTRACT FROM AUTHOR]

We study the continued fractions I 1 (τ) and I 2 (τ) of order sixteen by adopting the theory of modular functions. These functions are analogues of Rogers–Ramanujan continued fraction r (τ) with modularity and many interesting properties. Here we prove the modularities of I 1 (τ) and I 2 (τ) to find the relation with the generator of the field of modular functions on Γ 0 (1 6). Moreover we prove that the values 2 (I 1 (τ) 2 + 1 / I 1 (τ) 2) and 2 (I 2 (τ) 2 + 1 / I 2 (τ) 2) are algebraic integers for certain imaginary quadratic quantity τ. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study the quantities L⊕ (resp. L⊙) defined respectively as the maximal finite length of the β-fractional part appearing when one adds (resp. multiplies) two β-integers. We start by establishing some conditions for finiteness of L⊕ (resp.L⊙). Then we determine the exact value of L⊙ for the Tribonacci number. [ABSTRACT FROM AUTHOR]

The triplet (a, b, c) of positive integers is said to be compositum-feasible if there exist number fields K and L of degrees a and b, respectively, such that the degree of their compositum KL equals c. We determine all compositum-feasible triplets (a, b, c) satisfying a ≤ b and b ∈ {8, 9}. This extends the classification of compositum-feasible triplets started by Drungilas, Dubickas, and Smyth [5]. Moreover, we obtain several results related to triplets of the form (a, a, c). In particular, we prove that the triplet (n, n, n(n − 2)) is not compositum-feasible, provided that n ≥ 5 is an odd integer. [ABSTRACT FROM AUTHOR]

This article provides definitions and examples upon an integral element of unital commutative rings. An algebraic number is also treated as consequence of a concept of "integral". Definitions for an integral closure, an algebraic integer and a transcendental numbers [14], [1], [10] and [7] are included as well. As an application of an algebraic number, this article includes a formal proof of a ring extension of rational number field ℚ induced by substitution of an algebraic number to the polynomial ring of ℚ[x] turns to be a field. [ABSTRACT FROM AUTHOR]

The article offers an analysis on the distance between conjugate algebraic numbers in clusters. The estimation of discriminants and resultants is provided. It also discusses the normalized value of the modulus of the derivative at the conjugate root of polynomials which are present in the theorem of inequalities.

For any positive integer parameters a and b, Gurvich recently introduced a generalization mex of the standard minimum excludant mex = mex, along with a game NIM( a, b) that extends further Fraenkel's NIM = NIM( a, 1), which in its turn is a generalization of the classical Wythoff NIM = NIM(1, 1). It was shown that P-positions (the kernel) of NIM( a, b) are given by the following recursion: and conjectured that for all a, b the limits ℓ( a, b) = x( a, b)/ n exist and are irrational algebraic numbers. In this paper we prove that showing that $${\ell(a,b) = \frac{a}{r-1}}$$, where r > 1 is the Perron root of the polynomialwhenever a and b are coprime; furthermore, it is known that ℓ( ka, kb) = kℓ( a, b). In particular, $${\ell(a, 1) = \alpha_a = \frac{1}{2} (2 - a + \sqrt{a^2 + 4})}$$. In 1982, Fraenkel introduced the game NIM( a) = NIM( a, 1), obtained the above recursion and solved it explicitly getting $${x_n = \lfloor \alpha_a n \rfloor, \; y_n = x_n + an = \lfloor (\alpha_a + a) n \rfloor}$$. Here we provide a polynomial time algorithm based on the Perron-Frobenius theory solving game NIM( a, b), although we have no explicit formula for its kernel. [ABSTRACT FROM AUTHOR]

Abstract: We generalize to an arbitrary number field the result of Davenport and Schmidt about approximation to a real number and his square by rational numbers with the same denominator, as well as its p-adic analog due to Teulié. By an extension of Royʼs method, we prove that our new result is optimal. [Copyright &y& Elsevier]

Let P be a polynomial of degree d with integer coefficients such that P(0) ≠ 0. Assuming that P has no reciprocal factors we obtain a lower bound on the modulus of the smallest root of P in terms of its degree d, its Mahler measure M( P) and the number of roots of P lying outside the unit circle, say, k. We derive from this that all d roots of P must lie in the annulus R < | z| < R, where R = R( d, k, M( P)) and R = R( d, k, M( P)) are given explicitly. As an application, for non-reciprocal conjugate algebraic numbers α, α′ of degree d ≥ 2 and of Mahler's measure M( α), we prove the inequality $${|\alpha\alpha'-1|\,{ > }\,(12M(\alpha)^2 \log M(\alpha))^{-d}}$$. Some lower bounds on the moduli of the conjugates of a Pisot number are also given. In particular, it is shown that if α is a cubic Pisot number, then the disc | z| ≤ α + 0.1999 α contains no conjugates of α. Here the constant 0.1999 cannot be replaced by the constant 0.2. We also show that if α is a Pisot number of degree at least 4 and α′ is its conjugate, then | α α′ − 1| > (19 α). [ABSTRACT FROM AUTHOR]

We describe an algorithm for finding integer solutions of a system of simultaneous Pell equations whose effective estimates can be obtained using the theory of linear forms in the logarithms of algebraic numbers. We use Matveev’s estimate for forms in three logarithms. To decrease the resulting estimate, we use an iterative algorithm. At the end of the paper, results of the practical implementation of the proposed algorithm are given. [ABSTRACT FROM AUTHOR]

We prove the existence of modular polynomials for the lambda function and present an asymptotic formula for the maximum of the moduli of their coefficients. [ABSTRACT FROM AUTHOR]

We prove the existence of modular polynomials for the lambda function and present an asymptotic formula for the maximum of the moduli of their coefficients. [ABSTRACT FROM AUTHOR]

Abstract: Based on precomputed Sturm–Habicht sequences, discriminants and invariants, we classify, isolate with rational points, and compare the real roots of polynomials of degree up to 4. In particular, we express all isolating points as rational functions of the input polynomial coefficients. Although the roots are algebraic numbers and can be expressed by radicals, such representation involves some roots of complex numbers. This is inefficient, and hard to handle in applications in geometric computing and quantifier elimination. We also define rational isolating points between the roots of the quintic. We combine these results with a simple version of Rational Univariate Representation to isolate all common real roots of a bivariate system of rational polynomials of total degree ≤2 and to compute the multiplicity of these roots. We present our software within library synaps and perform experiments and comparisons with several public-domain implementations. Our package is 2–10 times faster than numerical methods and exact subdivision-based methods, including software with intrinsic filtering. [Copyright &y& Elsevier]

The article features the study conducted on the asymptotics of certain number-theoretic sums in Krasnoyarsk, Russia. Researchers solve the question of the asymptotics of a sum which is similar to σ is a quadriatic irrationality, wherein θ is a quadriatic irrationality such that θ + θ' ∈Z, which denotes the corresponding conjugate number. Presented are the formulas of asymptotics equations.

Abstract: Lind and Schmidt have shown that for certain ergodic -actions on a compact abelian group Γ, the homoclinic group H is isomorphic to the Pontryagin dual of Γ. Einsiedler and Schmidt extended these results and showed that Γ is a quotient of a locally compact ring modulo H. In this paper, we present a dynamical interpretation of if : it is a product of the stable group and the unstable group of Γ, under a suitable topology. As applications, we give a topological interpretation of the Pisot–Vijayaraghavan theorem and we link the results to tessellation theory. [Copyright &y& Elsevier]

We consider generalizations of Pólya’s theorem to the case of entire functions taking algebraic values at natural or integer points. [ABSTRACT FROM AUTHOR]

An upper bound for the index of a sublattice, which arises in relation to various versions of zero lemmas in the theory of linear forms in logarithms of algebraic numbers, in terms of the Hilbert polynomial is found. Simultaneously, a lower bound for the values of this polynomial is obtained. [ABSTRACT FROM AUTHOR]

The algebraic independence of $$ e^{{\theta _{1} }} , \ldots ,e^{{\theta _{s} }} $$ is proved, where θ1, . . . , θs are certain gap series or power series of algebraic numbers, or certain transcendental continued fractions with algebraic elements. [ABSTRACT FROM AUTHOR]

We prove a special case of Siegel's conjecture concerning the representability of E-functions in the form of polynomials in hypergeometric functions. We prove several assertions (formulated earlier by A. B. Shidlovskii) about the transcendence and linear independence of values of E-functions. [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]

Abstract: In this paper, we study a sweeping algorithm for computing the arrangement of a set of quadrics in . We define a “trapezoidal” decomposition in the sweeping plane, and we study the evolution of this subdivision during the sweep. A key point of this algorithm is the manipulation of algebraic numbers. In this perspective, we put a large emphasis on the use of algebraic tools, needed to compute the arrangement, including Sturm sequences and Rational Univariate Representation of the roots of a multivariate polynomial system. [Copyright &y& Elsevier]

The reciprocal square root calculation α=1/√ of x is very common in scientific computations. Having a correctly rounded implementation of it is of great importance in producing numerically predictable code among today''s heterogenous computing environment. Existing results suggest that to get the correctly rounded α in a floating point number system with p significant bits, we may have to compute up to 3p+1 leading bits of α. However, numerical evidence indicates the actual number may be as small as 2p plus a few more bits. This paper attempts to bridge the gap by showing that this is indeed true, assuming the abc conjecture which is widely purported to hold. (But our results do not tell exactly how many more bits beyond the 2p bits, due to the fact that the constants involved in the conjecture are ineffective.) Along the way, rough bounds which are comparable to the existing ones are also proven. The technique used here is a combination of the classical Liouville''s estimation and contemporary number theory. [Copyright &y& Elsevier]

By an automorphism of a topological group G we mean an isomorphism of G onto itself which is also a homeomorphism. In this article, we study the automorphism group Aut (G) of a dense subgroup G of R n , n ≥ 1. We show that Aut (G) can be naturally identified with the subgroup Φ (G) = { A ∈ G L (n , R) : G ⋅ A = G } of the group G L (n , R) of all non-degenerated (n × n) -matrices with real coefficients, where G ⋅ A = { g ⋅ A : g ∈ G }. We describe Φ (G) for many dense subgroups G of either R or R 2. We consider also an inverse problem of which symmetric subgroups of G L (n , R) can be realized as Φ (G) for some dense subgroup G of R n. For n ≥ 2 , we show that any subgroup H of G L (n , R) satisfying S O (n , R) ⊆ H ⊊ G L (n , R) cannot be realized in this way. (Here S O (n , R) denotes the special orthogonal group of dimension n.) The realization problem is quite non-trivial even in the one-dimensional case and has deep connections to number theory. [ABSTRACT FROM AUTHOR]

The elementary symmetric functions play a crucial role in the study of zeros of non-zero polynomials in C [ x ] , and the problem of finding zeros in Q [ x ] leads to the definition of algebraic and transcendental numbers. Recently, Marques studied the set of algebraic numbers in the form P (T) Q (T) . In this paper, we generalize this result by showing the existence of algebraic numbers which can be written in the form P 1 (T) Q 1 (T) ⋯ P n (T) Q n (T) for some transcendental number T, where P 1 , ... , P n , Q 1 , ... , Q n are prescribed, non-constant polynomials in Q [ x ] (under weak conditions). More generally, our result generalizes results on the arithmetic nature of z w when z and w are transcendental. [ABSTRACT FROM AUTHOR]