Simultaneous approximation of et and ℘(t)

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.

We establish various new results on a problem proposed by Mahler [Some suggestions for further research. Bull. Aust. Math. Soc.29 (1984), 101–108] concerning rational approximation to fractal sets by rational numbers inside and outside the set in question. Some of them provide a natural continuation and improvement of recent results of Broderick, Fishman and Reich, and Fishman and Simmons. A key feature is that many of our new results apply to more general, multi-dimensional fractal sets and require only mild assumptions on the iterated function system. Moreover, we provide a non-trivial lower bound for the distance of a rational number $p/q$ outside the Cantor middle-third set $C$ to the set $C$, in terms of the denominator $q$. We further discuss patterns of rational numbers in fractal sets. We highlight two of them: firstly, an upper bound for the number of rational (algebraic) numbers in a fractal set up to a given height (and degree) for a wide class of fractal sets; and secondly, we find properties of the denominator structure of rational points in ‘missing-digit’ Cantor sets, generalizing claims of Nagy and Bloshchitsyn.

CHAPTER TEN - ROOTS OF POLYNOMIALS

Throughout these notes V will be a finite dimensional K vector space and I will denote the identity operator. For all our applications there is no loss in specializing K to be the field of rational numbers, but at some point in our development we will need to assume that K is the field of complex numbers.

We present a survey of complexity results for constraint satisfaction problems (CSPs) over the integers, the rationals, the reals, and the complex numbers. Examples of such problems are feasibility of linear programs, integer linear programming, the max-atoms problem, Hilbert's tenth problem, and many more. Our particular focus is to identify those CSPs that can be solved in polynomial time, and to distinguish them from CSPs that are NP-hard. A very helpful tool for obtaining complexity classifications in this context is the concept of a polymorphism from universal algebra.

We will show that this construction is impossible if a straightedge and compass are the only tools which we may use. We precede the proof with a couple of definitions and facts which will be needed. A complex number is algebraic if it is a zero of a polynomial with integer coefficients. Numbers which are not algebraic are called transcendental. The set of all algebraic numbers is a field. This implies, in particular, that for any algebraic number a and any natural number k the number ak iS also algebraic. Recall also that each constructible number, i.e., a number which can be constructed using a straightedge and compass only, is algebraic. A crucial result for us is the following theorem due to Lindemann. If x 7s 0 is an algebraic number, then ex is a transcendental number. We refer to [2, 3] for unexplained terminology and details.

Let K be a biquadratic field. M.-N. Gras and F. Tanoe gave a necessary and sufficient condition that K is monogenic by using a diophantine equation of degree 4 [3]. We consider algebraic extension fields of higher degree. Let F be a Galois extension field over the rationals % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf % gDOjdaryqr1ngBPrginfgDObcv39gaiuaacqWFAesuaaa!40A2! $$\mathbb{Q}$$ whose Galois group is 2-elementary abelian. Then we shall prove that F of degree % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4waiaadA % eacaGG6aWefv3ySLgznfgDOjdaryqr1ngBPrginfgDObcv39gaiuaa % cqWFAesucaGGDbGaeyyzImRaaGioaaaa!4673! $$[F:\mathbb{Q}] \geqq 8$$ , is monogenic if and only if % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraiabg2 % da9mrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbacfaGae8Ng % HeLaaiikamaakaaabaGaeyOeI0IaaGymaaWcbeaakiaacYcadaGcaa % qaaiabgkHiTiaaikdaaSqabaGccaGGSaWaaOaaaeaacqGHsislcaaI % ZaaaleqaaOGaaiykaiabg2da9iab-PrirjaacIcacqaH2oGEdaWgaa % WcbaGaaGOmaiaaisdaaeqaaOGaaiykaaaa!5173! $$F = \mathbb{Q}(\sqrt { - 1} ,\sqrt { - 2} ,\sqrt { - 3} ) = \mathbb{Q}(\zeta _{24} )$$ under a suitable condition for the case of degree 8.

For the time being the objects of arithmetic are the whole numbers, 0, ± 1, ± 2,... which can be combined by addition, subtraction, multiplication and division (not always) to form integers. Higher arithmetic uses methods of investigation analogous to those of real or complex numbers. Moreover it also uses analytic methods which belong to other areas of mathematics, such as infinitesimal calculus and complex function theory, in the derivation of its theorems. Since these will also be discussed in the latter part of this book, we will assume as known the totality of complex numbers, a number domain, in which the four types of operations (except division by 0) can be carried out unrestrictedly. The complex domain is usually developed more precisely in the elements of algebra or of differential calculus. In this domain the number 1 is distinguished as the one which satisfies the equation $$1 \cdot a = a$$ for each number a. All successive integers are obtained by the process of addition and subtraction from the number 1, and if the process of division is then carried out the set of rational numbers is obtained as the totality of quotients of integers. Later, from §21 on, the concept of “integer” will be subjected to an essential extension.KeywordsResidue ClassEssential ExtensionGreatest Common DivisorPrimitive PolynomialIntegral PolynomialThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

CHAPTER ELEVEN - ROOTS OF POLYNOMIALS

The number λq = 2cosπ/q, q ∈ N, q ≥ 3, appears in the study of Hecke groups which are Fuchsian groups, and in the study of regular polyhedra. There are many results about the minimal polynomial of this algebraic number and in some of these methods, the minimal polynomials of several algebraic numbers are used. Here we obtain the minimal polynomial of one of those numbers, cos(2π/n), over the field of rationals by means of the better known Chebycheff polynomials for odd q and give some of their properties. We calculated this minimal polynomial for n ∈ N by using the Maple language and classifying the numbers n ∈ N into different classes.

It is known that the transcendental and rational numbers in the unit interval are not binomial numbers. In this article we will show that the algebraic integers of degree 2 are not binomial numbers either. Therefore two shift invariant measures u ( s ) , u ( r ) u(s),u(r) with r r being an algebraic integer of degree 2 in the unit interval are topologically equivalent if and only if s = r s = r or s = 1 − r s = 1 - r . We also show that for each positive integer n > 2 n{\text { > 2}} , there are algebraic integers and fractionals of degree n n in the unit interval that are binomial numbers.

This paper formulates general conjectures relating the structure of the 2-classgroup C 2 ( d p ) {C_2}(dp) associated to Q ( d p ) {\mathbf {Q}}(\sqrt {dp} ) to the splitting of the ideal ( p ) in certain algebraic number fields. Here d ≢ 2 d\,\nequiv \,2 ( mod 4 ) \pmod 4 is a fixed integer and p varies over primes. The conjectures assert that there exists an algebraic number field Ω j ( d ) {\Omega _j}(d) such that the Artin symbol [ ( Ω j ( d ) / Q ) / ( p ) ] [({\Omega _j}(d)/{\mathbf {Q}})/(p)] determines the first j 2-invariants of the group C 2 ( d p ) {C_2}(dp) , i.e. it determines C 2 ( d p ) / C 2 ( d p ) 2 j {C_2}(dp)/{C_2}{(dp)^{{2^j}}} . These conjectures imply that the set of primes p for which C 2 ( d p ) {C_2}(dp) has a given set of 2-invariants has a natural density which is a rational number. Existing results prove the conjectures whenever j = 1 j = 1 or 2 and also for an infinite set of d with j = 3 j = 3 . The smallest open case is j = 3 j = 3 , d = − 21 d = - 21 . This paper presents evidence concerning these conjectures for d = − 4 d = - 4 , 8 and − 21 - 21 . Numerical evidence is given that Ω 3 ( − 21 ) {\Omega _3}( - 21) exists, and that natural densities which are rational numbers exist for the sets of primes with 2 j | h ( d p ) {2^j}|h(dp) for d = − 4 d = - 4 and 8, for 1 ⩽ j ⩽ 7 1 \leqslant j \leqslant 7 . A search for the hypothetical field Ω 4 ( − 4 ) {\Omega _4}( - 4) ruled out the simplest candidate fields: Ω 4 ( − 4 ) {\Omega _4}( - 4) is not a normal extension of Q of degree 16 ramifying only at (2).

To perform the numerous tasks of a computer algebra system, a multitude of simple and complex data types, operators and statements are necessary. Important elementary data types are the various numerical data types like floating point numbers, complex numbers, rational numbers and integers of any length as well as identifiers and character strings. Additionally indexed identifiers are necessary to enable access to tables, arrays, lists and expression sequences. To control, for instance, the execution of statements, it must be possible to evaluate expressions to boolean values. In this context, the boolean constants TRUE and FALSE are important. Further available constants are E, EULER and PI. More complex data types allow objects to be combined to form a new object. These data types include lists, sets, tables and arrays and domains.

Out of the box, Common Lisp is exceptionally well suited to numerical mathematical calculations. Integers are not limited to a specific size (see Recipe 4-1). It comes with rational numbers so you can do arithmetic with unlimited precision (see Recipe 4-5). And it has complex numbers (see Recipe 4-8).

A complex number [Formula: see text] is said to be non-free if the subgroup of [Formula: see text] generated by [Formula: see text] is not a free group of rank 2. In this case, the number [Formula: see text] is called a relation number, and it has been a long standing problem to determine the relation numbers. In this paper, we characterize the relation numbers by establishing the equivalence between [Formula: see text] being a relation number and [Formula: see text] being a root of a ‘generalized Chebyshev polynomial’. The generalized Chebyshev polynomials of degree [Formula: see text] are given by a sequence of [Formula: see text] integers [Formula: see text] using the usual recursive formula, and thereby can be studied systematically using continuants and continued fractions. Such formulation, then, enables us to prove that, the question whether a given number [Formula: see text] is a relation number of [Formula: see text]-degree [Formula: see text] can be answered by checking only finitely many generalized Chebyshev polynomials. Based on these theorems, we design an algorithm deciding any given number is a relation number with minimal degree [Formula: see text]. With its computer implementation we provide a few sample examples, with a particular emphasis on the well-known conjecture that every rational number in the interval [Formula: see text] is a relation number.