КЛАССИФИКАЦИЯ ЧИСТО-ВЕЩЕСТВЕННЫХ АЛГЕБРАИЧЕСКИХ ИРРАЦИОНАЛЬНОСТЕЙ

Convolution shift, c-orthogonality preserving maps, and the Laguerre polynomials

Для каждого $n \ge 3$ ранее были построены три неэквивалентных многочлена $f \in \mathbb{Q}[x]$ степени $n$, для которых $\sqrt{f}$ имеет периодическое разложение в непрерывную дробь в поле $\mathbb{Q}((x))$. В работе для каждого $n \ge 5$ найдены по два новых многочлена $f \in K[x]$ степени $n$, которые определены над полем $K$, $[K : \mathbb{Q}] = [(n-1)/2]$, и для которых $\sqrt{f}$ имеет периодическое разложение в непрерывную дробь в поле $K((x))$.

The obvious way to compute the continued fraction of a real number α > 1 is to compute a very accurate numerical approximation of α, and then to iterate the well-known truncate-and-invert step which computes the next partial quotient a = bαc and the next complete quotient α′ = 1/(α − a). This method is called the basic method. In the course of this process precision is lost, and one has to take precautions to stop before the partial quotients become incorrect. Lehmer [6] gives a safe stopping-criterion, and a trick to reduce the amount of multi-length arithmetic. Schonhage [12] describes an algorithm for computing the greatest common divisor of u and v and the related continued fraction expansion of u/v in O(n log n log log n) steps if both u and v do not exceed 2. A disadvantage of this approach is that if we wish to extend the list of partial quotients computed from an initial approximation of α, we have to compute a more accurate initial approximation of α, compute the new complete quotient using this new approximation and the partial quotients already computed from the old approximation, and then extend the list of partial quotients using that new complete quotient (we notice that Shiu in [13, p.1312] incorrectly states that all the previous calculations have to be repeated, but the partial quotients computed so far don’t have to be recomputed). Bombieri and Van der Poorten [1], and Shiu [13] have recently proposed a remedy for this problem. They give a formula for computing a rational approximation of the next complete convergent from the first n partial quotients. From that complete convergent about n new partial quotients can be computed. So each step gives an approximate doubling of the number of partial quotients. To start the method, a few partial quotients have to be computed with the basic or indirect method. In [1] this method is proposed for algebraic numbers (which are zeros of polynomials) of degree ≥ 3, whereas Shiu also applies it to more general numbers, namely to transcendental numbers that can be defined as the zero of a function for which the logarithmic derivative at a rational point can be computed with arbitrary precision. This includes numbers like π, log π, and log 2. For each of thirteen different numbers, Shiu computes 10000 partial quotients. Their frequency distributions are compared with the one which almost all numbers should obey, according to the Khintchine– Levy theory [3, 7]. No significant deviations from this theory are reported. Shiu calls his method the direct method. Curiously, Shiu does not refer to what we would call the polynomial method for algebraic numbers [2, 5, 11] of degree ≥ 3, which computes the partial quotients of α using only the coefficients of its defining polynomial. Moreover, Shiu gives neither implementational

The Minimal Euclidean Norm of an Algebraic Number Is Effectively Computable

We present a formalization in ACL2(r) of three proofs originally done by Cantor. The first two are different proofs of the non-denumerability of the reals. The first, which was described by Cantor in 1874, relies on the completeness of the real numbers, in the form that any infinite chain of closed, bounded intervals has a non-empty intersection. The second proof uses Cantor’s celebrated diagonalization argument, which did not appear until 1891. The third proof is of the existence of real transcendental (i.e., non-algebraic) numbers. It also appeared in Cantor’s 1874 paper, as a corollary to the non-denumerability of the reals. What Cantor ingeniously showed is that the algebraic numbers are denumerable, so every open interval must contain at least one transcendental number.

Limits of characteristic sequences of integer-valued polynomials on homogeneous sets

A vector m = ( m 1 , … , m n ) ∈ Z n ∖ { 0 } m = ({m_1}, \ldots ,{m_n}) \in {{\mathbf {Z}}^n}\backslash \{ 0\} is called an integer relation for the real numbers α 1 , … , α n {\alpha _1}, \ldots ,{\alpha _n} , if ∑ α i m i = 0 \sum {\alpha _i}{m_i} = 0 holds. We present an algorithm that, when given algebraic numbers α 1 , … , α n {\alpha _1}, \ldots ,{\alpha _n} and a parameter ε \varepsilon , either finds an integer relation for α 1 , … , α n {\alpha _1}, \ldots ,{\alpha _n} or proves that no relation of Euclidean length shorter than 1 / ε 1/\varepsilon exists. Each algebraic number is assumed to be given by its minimal polynomial and by a sufficiently precise rational approximation. Our algorithm uses the Lenstra-Lenstra-Lovász lattice basis reduction technique. It performs \[ poly ( log ⁡ 1 / ε , n , log ⁡ max i height ( α i ) , [ Q ( α 1 , … , α n ) : Q ] ) {\operatorname {poly}}\left ( {\log 1/\varepsilon ,n,\log \max \limits _i {\text {height}}({\alpha _i}),[{\mathbf {Q}}({\alpha _1}, \ldots ,{\alpha _n}):{\mathbf {Q}}]} \right ) \] bit operations. The straightforward algorithm that works with a primitive element of the field extension Q ( α 1 , … , α n ) {\mathbf {Q}}({\alpha _1}, \ldots ,{\alpha _n}) of Q would take \[ poly ( n , log ⁡ max i height ( α i ) , ∏ i = 1 n degree ( α i ) ) {\operatorname {poly}}\left ( {n,\log \max \limits _i {\text {height}}({\alpha _i}),\prod \limits _{i = 1}^n {{\text {degree}}({\alpha _i})} } \right ) \] bit operations. In order to prove the correctness of the algorithm, we show a lower bound for | ∑ α 1 m i | \left | {\sum {\alpha _1}{m_i}} \right | if m is not an integer relation for α 1 , … , α n {\alpha _1}, \ldots ,{\alpha _n} , which may be interesting in its own right.

The development of bounds on the complex effective conductivity tensor σ* (that relates the average current to the average electric field in a multicomponent composite) has been hindered by lack of a suitable continued-fraction representation for σ*. Here a new field equation recursion method is developed which gives an expression for σ* as a continued fraction of a novel form incorporating as coefficients the component conductivities and a set of fundamental geometric parameters reflecting the composite geometry. A hierarchy of field equations is set up such that the solutions of the (j+1)th-order equation generate the solutions of thejth-order equation. Consequently the effective tensor Ω(j) associated with thejth-order field equation is expressible as a fractional linear matrix transformation of Ω(j+1). These transformations combine to form the continued fraction expansion for σ*=Ω(0) which is exploited in the following paper, Part II, to obtain bounds: crude bounds on Ω(j), forj≧1, give narrow bounds on σ*. The continued fraction is a generalization to multivariate functions of the continued fraction expansion of single variable Stieltjes functions that proved important in the development of the theory of Páde approximants, asymptotic analysis, and the theory of orthogonal polynomials in the last century. The results extend to other transport problems, including conduction in polycrystalline media, the viscoelasticity of composites, and the response of multicomponent, multiterminal linear electrical networks.

Let α and β be any two real numbers and let be the Jacobi polynomial sequences. For any non-zero real number a, is an orthogonal polynomial sequence with respect to convolution if and only if either (i) b = 1, α = 0 and β + 1 is not equal to a negative integer or (ii) b = — 1, β = 0 and α + 1 is not equal to a negative integer.

On matrix Hurwitz type polynomials and their interrelations to Stieltjes positive definite sequences and orthogonal matrix polynomials

Abstract. We study the computational complexity of linear programs with coefficients that are real algebraic numbers under a Turing machine model of computation. After reviewing a method for exact representation of algebraic numbers under the Turing model, we show that the fundamental tasks of comparison and arithmetic can be performed in polynomial time. Our technique for establishing polynomial-time algorithms for comparison and arithmetic is distinct from the usual resultant-based approaches, and has the advantage that it provides a natural framework for analysis of the complexity of computational tasks, such as Gaussian elimination, that involve a sequence of arithmetic operations. Our main contribution is to show that a variant of the ellipsoid method can be used to solve linear programming in time polynomial in the encoding size of the problem coefficients and the degree of any algebraic extension that contains those coefficients.

A vector m=(m1,...,m n ) ∈ Zn {0} is called an integer relation for the real numbers α1,...,α n , if Σα i m i =0 holds. We present an algorithm that when given algebraic numbers α1,...,α n and a parameter ɛ either finds an integer relation for α1,...,α n or proves that no relation of euclidean length shorter than 1/ɛ exists. Each algebraic number is assumed to be given by its minimal polynomial and by a rational approximation precise enough to separate it from its conjugates.

Continued fractions with multiple limits

We prove that if f(x) = YlkZo akxk is a polynomial with no cyclotomic factors whose coefficients satisfy ak = 1 mod 2 for 0 2. We also characterize the polynomials that appear as the noncyclotomic part of a polynomial whose coefficients satisfy ak = 1 mod p for each ft, for a fixed prime p. Last, we prove that the smallest Pisot number whose minimal polynomial has odd coefficients is a limit point, from both sides, of Salem [19] numbers whose minimal polynomials have coef ficients in {?1,1}.

The modular group k is the quotient group PSL2(Z) = SL,(Z)/{I we refer to these as reduced strings. We explained this free product structure in terms of the action of the modular group on the irrationals. In this note we describe the action on the rationals; this can be viewed as a way of describing the inverse of the Euclidean algorithm. The group SL2(Z) acts via linear transformations on R2 as column vectors and this gives an action of .A' via linear fractional transformations on the projective line P'(R), the real numbers together with m. We may also view P'(R) as the slopes of non-zero vectors, that is, the equivalence classes of R2 -(:)induced by non-zero scalar multiplication; the equivalence class of the vector e = ( i ) is , ,