Recovery of algebraic numbers from their p-adic approximations

where the Li(x) are homogeneous polynomials of degree 1 with coefficients which are algebraic numbers. One of the most interesting cases arises when the following two conditions are satisfied. (1) The Li(x) never vanish simultaneously at any of V. (2) Under the mapping a of V again becomes a of V. Generally speaking the only points which interest us are those which can be expressed with coordinates which are algebraic numbers. = (to, I, * * * n is such a point, the number field generated by the ratios ti/tj will be the smallest field in which is rational. The degree of this field over the rational numbers, we shall call the degree of rationality of P. Suppose now that conditions (1) and (2) are satisfied, then by iterating the mapping, a given of V will generate an infinite sequence = Po, P1, P2, *** of points on the variety. This sequence will consist either of distinct points or else there will be repetitions. In the latter case the sequence ultimately becomes periodic, and we shall describe this situation by saying P is an exceptional point. One of the results established can now be stated, namely, If we exclude the case in which the mapping is linear, then there will be at most a finite number of exceptional points with a given degree of rationality. We can illustrate this result by means of the well known Weierstrass function P(Z1 g2, g3), where g2 and g3 are algebraic numbers. A complex number 'a' is called a point if for some integer n, which is not zero, na is a period. The values of p(z) at the division points are called values, and it is easy to see that these are algebraic numbers. By comparing the elliptic function with the exponential function, we see that these division values are in some respects analogous to roots of unity, and it is therefore natural to inquire how they are distributed among the number fields. A special case of the general theorem stated above, shows that if we limit ourselves to number fields whose absolute degrees are bounded by some given integer, then amongst all these fields there will be only a finite number of division values. These facts are consequences of Theorem 2 which is an inequality proved under very general conditions. This inequality has another application (not discussed here), namely 167

Let $\{ {x_i } \}$ be any sequence approximating an algebraic number $\alpha $ of degree r, and let $x_{i + 1} = \varphi (x_i ,x_{i - 1} , \cdots ,x_{i - d + 1} )$, for some rational function $\varphi $ with integral coefficients. Let M denote the number of multiplications or divisions needed to compute $\varphi $ and let $\bar M$ denote the number of multiplications or divisions, except by constants, needed to compute $\varphi $. Define the multiplicative efficiency measure of $\{ {x_i } \}$ as $E = {{(\log _2 p)} / M}$ or as $\bar E = {{(\log _2 p)} / {\bar M}}$, where p is the order of convergence of $\{ {x_i } \}$. Kung [1] showed that $\bar E \leqq 1$ or equivalently, $\bar M \geqq \log _2 p$. In this paper we show that (i) $\bar M \geqq \log _2 [r(\lceil p \rceil - 1) + 1] - 1$; (ii) if $E = 1$ then $\alpha $ is a rational number; (iii) if $\bar E = 1$ then $\alpha $ is a rational or quadratic irrational number. This settles the question of when the multiplicative efficiency E or $\bar E$ achieves its optimal value of unity. Also, as a consequence of result (i), we show that the maximal efficiency $\bar E$ achievable by algebraic numbers of degree r drops at least as $O[(\log r)^{ - 1} ]$, provided that we only consider sequences $\{ {x_i } \}$ of bounded order of convergence.

In this paper, to identify irrational numbers while appropriately emphasizing basic functions. All rational numbers can be expressed algebraically, but not all irrational numbers. Rational, irrational, algebraic numbers and transcendental numbers are mostly concerned with irrational numbers using some elementary functions.

Computational complexity of quantifier-free negationless theory of field of rational numbers

Around the late 1970s, Rohrlich made a conjecture about multiplicative algebraic relations among the special values of the Γ- function. Later, Lang generalized the Rohrlich conjecture to polynomial algebraic relations among special values of the gamma function. In 2009, Gun et al. (J. Number Theory 129 (2009), no. 8, 1858–1873) formulated a variant of this conjecture of Rohrlich and a variant of the conjecture of Lang that deals with the linear independence of the values at non-integeral rational numbers of the logarithm of the gamma function over the field of rationals and algebraic numbers, respectively. In this direction, they proved a set of interesting results for the case of primes and their powers over the field of rationals. Further for the case of prime powers, they have extended their results assuming the Schanuel's conjecture. In this article, we improve their results without assuming Schanuel's conjecture. Further we provide counter examples to these variants of conjectures of Rohrlich and Lang for an infinite class of integers having at least two prime factors satisfying certain conditions.

Companions of the field of rational numbers and a real-closed algebraic expansion of the field of rational numbers are studied. The description of existentially closed companions of a real-closed algebraic expansion of a field of rational numbers refers to the field of study of classical algebraic structures. The general theory of companions and existentially closed companions, built on the basis of Fraisse's classes in the works of A.T. Nurtazin, is included in the classical field of existentially closed theories in model theory. The basic concept of a companion: two models of the same signature are called companions if for any finite submodel of one of them, there is an isomorphic finite submodel in the other. This approach, applied to specific classical structures and their theories, provides new tools for the study of these objects. The study of the companion class of rational and algebraic real number fields reveals companion fields containing transcendental and possibly algebraic elements with special properties of polynomials defining these elements.

(1) ak+n = c1ak+n−1 + . . .+ cnak (k = 0, 1, 2, . . .), where a0, . . . , an−1 are not all zero and c1, . . . , cn are nonnegative integers with cn 6= 0. Put (2) Φ(X) = X − c1Xn−1 − . . .− cn. In what follows, Q and Q denote the fields of rational and algebraic numbers respectively. In 1929, Mahler [4] proved the following theorem: Let {ak}k≥0 be a linear recurrence satisfying (1). Suppose that Φ(X) is irreducible over Q and the roots %1, . . . , %n of Φ(X) satisfy %1 > max{1, |%2|, . . . . . . , |%n|}. If α is an algebraic number with 0 < |α| < 1, then the number ∑∞ k=0 α ak is transcendental. In this paper, we establish two theorems on the algebraic independence of the values of power series generated by linear recurrences with conditions on Φ(X) weaker than those of Mahler (see Remark 1 below). Let {ak}k≥0 and {bk}k≥0 be linear recurrences satisfying (1). We write {ak}k≥0 ∼ {bk}k≥0 if there is a nonnegative integer l such that ak = bk+l (0 ≤ k ≤ n− 1) or bk = ak+l (0 ≤ k ≤ n− 1). Then ∼ is an equivalence relation. Its negation is written as {ak}k≥0 6∼ {bk}k≥0. We denote by f (l)(z) the lth derivative of a function f(z). Theorem 1. Let {a k }k≥0 (i = 1, . . . , s) be linear recurrences satisfying (1). Suppose that Φ(±1) 6= 0 and the ratio of any pair of distinct roots of Φ(X) is not a root of unity. Let

The dissertation is about algorithms of algebraic numbers and related theoretical questions. An important aspect of the discussed algorithms is that they calculate exactly, not numerically. In this way we avoid the usual problem of numerical calculations, the rounding errors, but the exact algorithms are usually slower, and they often suffer from coefficient explosion. Avoiding this problem and maintaining the running time (i.e. ensuring polynomial time) is an important aspect of exact algorithms. The dissertation has two major parts. The first half deals with the computational complexity of operations and algorithms involving algebraic numbers. We calculate the running time of the main operations in algebraic number fields (using some existing results). We also analyse some special operations like comparison between real algebraic numbers and integer rounding. These results enable us to analyse two algorithms, the Bareiss algorithm (a generalisation of the Gaussian elimination) and the LLL algorithm (a lattice reduction algorithm) on algebraic numbers using exact arithmetic, and analyse their complexity. Their behaviour is well-known for integers, but we extend them to algebraic integers, and prove that their running time remains polynomial. The second half of the dissertation is about special algebraic numbers: roots of integer polynomials whose all roots are outside of the complex unit circle. Such polynomials are called expansive polinomials. We examine the question how to decide whether a polynomial with integer coefficients is expansive (without calculating its roots). There are some existing methods for this, but we introduce a new method that is guaranteed to avoid coefficient explosion. Then we examine how close the roots of an expansive integer polynomial can be to the unit circle, and give lower bounds on this distance in terms of the parameters of the polynomial. Then we construct a family of expansive polynomials that have roots very close to the unit circle, thus showing that the lower bounds are almost sharp.

Let n be a positive integer. Let ξ be an algebraic real number of degree greater than n. It follows from a deep result of W. M. Schmidt that, for every positive real number e, there are infinitely many algebraic numbers α of degree at most n such that |ξ−α| < H(α)−n−1+e, where H(α) denotes the naive height of α. We sharpen this result by replacing e by a function H 7→ e(H) that tends to zero when H tends to infinity. We make a similar improvement for the approximation to algebraic numbers by algebraic integers, as well as for an inhomogeneous approximation problem.

We derive an algorithm based on the ellipsoid method that solves linear programs whose coefficients are real algebraic numbers. By defining the encoding size of an algebraic number to be the bit size of the coefficients of its minimal polynomial, we prove the algorithm runs in time polynomial in the dimension of the problem, the encoding size of the input coefficients, and the degree of any algebraic extension which contains the input coefficients. This bound holds even if all input and arithmetic is performed symbolically, using rational numbers only.

In this thesis, we study the problem of simultaneous approximation to a fixed family of real and p-adic numbers by roots of integer polynomials of restricted type. The method that we use for this purpose was developed by H. DAVENPORT and W.M. SCHMIDT in their study of approximation to real numbers by algebraic integers. This method based on Mahler's Duality requires to study the dual problem of approximation to successive powers of these numbers by rational numbers with the same denominators. Dirichlet's Box Principle provides estimates for such approximations but one can do better. In this thesis we establish constraints on how much better one can do when dealing with the numbers and their squares. We also construct examples showing that at least in some instances these constraints are optimal. Going back to the original problem, we obtain estimates for simultaneous approximation to real and p-adic numbers by roots of integer polynomials of degree 3 or 4 with fixed coefficients in degree ≥ 3. In the case of a single real number (and no p-adic numbers), we extend work of D. Roy by showing that the square of the golden ratio is the optimal exponent of approximation by algebraic numbers of degree 4 with bounded denominator and trace.

The set of real numbers and the set of complex numbers have the power of continuum. Among these numbers, those which are ``interesting'', which appear ``naturally'', which deserve our attention, form a countable set. Starting from this point of view we are interested in the periods as defined by M.~Kontsevich and D.~Zagier. We give the state of the art on the question of the arithmetic nature of these numbers: to decide whether a period is a rational number, an irrational algebraic number or else a transcendental number is the object of a few theorems and of many conjectures. We also consider the approximation of such numbers by rational or algebraic numbers.

Contrary to popular misconception, the question in the title is far from simple. It involves sets of numbers on the first level, sets of sets of numbers on the second level, and so on, endlessly. The infinite hierarchy of the levels involved distinguishes the concept of "definable number" from such notions as "natural number", "rational number", "algebraic number", "computable number" etc.

Let $r, \,m$ be positive integers. Let $x$ be a rational number with $0 \le x <1$. Consider $\Phi_s(x,z) =\displaystyle\sum_{k=0}^{\infty}\frac{z^{k+1}}{{(k+x+1)}^s}$ the $s$-th Lerch function with $s=1, 2, \cdots, r$. When $x=0$, this is a polylogarithmic function. Let $\alpha_1, \cdots, \alpha_m$ be pairwise distinct algebraic numbers of arbitrary degree over the rational number field, with $0<|\alpha_j|<1 \,\,\,(1\leq j \leq m)$. In this article, we show a criterion for the linear independence, over an algebraic number field containing $\mathbb{Q}(\alpha_1, \cdots, \alpha_m)$, of all the $rm+1$ numbers : $\Phi_1(x,\alpha_1)$, $\Phi_2(x,\alpha_1), $ $\cdots , \Phi_r(x,\alpha_1)$, $\Phi_1(x,\alpha_2)$, $\Phi_2(x,\alpha_2), $ $\cdots , \Phi_r(x,\alpha_2), \cdots, \cdots, \Phi_1(x,\alpha_m)$, $\Phi_2(x,\alpha_m)$, $\cdots , \Phi_r(x,\alpha_m)$ and $1$. This is the first result that gives a sufficient condition for the linear independence of values of the Lerch functions at several distinct algebraic points, not necessarily lying in the rational number field nor in quadratic imaginary fields. We give a complete proof with refinements and quantitative statements of the main theorem announced in [10], together with a proof in detail on the non-vanishing Wronskian of Hermite type.

Chapter 54 - David Hilbert, report on algebraic number fields (‘Zahlbericht’) (1897)