Algebraic Integer Of Degree Research Articles

Klaus Friedrich Roth, 1925-2015

Klaus Friedrich Roth, 1925-2015

The beta-transformation's companion map for Pisot or Salem numbers and their periodic orbits

ABSTRACTThe β-transformation of the unit interval is defined by Tβ(x) ≔ βx (mod 1). Its eventually periodic points are a subset of [0, 1] intersected with the field extension . If β > 1 is an algebraic integer of degree d > 1, then is a -vector space isomorphic to ; therefore, the intersection of [0, 1] with is isomorphic to a domain in . The transformation from this domain which is conjugate to the β-transformation is called the companion map, given its connection to the companion matrix of β's minimal polynomial. The companion map and the proposed notation provide a natural setting to reformulate a classic result concerning the set of periodic points of the β-transformation for numbers. It also allows to visualize orbits in a d-dimensional space. Finally, we refer connections with arithmetic codings and symbolic representations of hyperbolic toral automorphisms.

Discriminants of 𝔖n-orders

Let [Formula: see text] be an algebraic integer of degree [Formula: see text]. Let [Formula: see text] be the [Formula: see text] complex conjugate of [Formula: see text]. Assume that the Galois group [Formula: see text] is isomorphic to the symmetric group [Formula: see text]. We give a [Formula: see text]-basis and the discriminant of the order [Formula: see text]. We end up with an open question showing that this problem seems much harder when we assume that [Formula: see text] is already Galois or even cyclic.

On a conjecture of Schinzel and Zassenhaus

A. Schinzel and H. Zassenhaus had the following conjecture regarding algebraic inte- gers: If α � 0 is an algebraic integer of degree n which is not a root of unity, then there exists a constant c > 0 such that |α | 1+ c , where |α | = max1in |αi|, α1 = α and α2,···,αn are the conjugates of α . We give some partial solutions to this conjecture in this paper via spectral properties.

On ℤ-Modules of Algebraic Integers

Abstract. Let q be an algebraic integer of degree d ≥ 2. Consider the rank of the multiplicative subgroup of ℂ* generated by the conjugates of q. We say q is of full rank if either the rank is d − 1 and q has norm ±1, or the rank is d. In this paper we study some properties of ℤ[q] where q is an algebraic integer of full rank. The special cases of when q is a Pisot number and when q is a Pisot-cyclotomic number are also studied. There are four main results.(1)If q is an algebraic integer of full rank and n is a fixed positive integer, then there are only finitely many m such that disc `ℤ[qm]´ = disc `ℤ[qn]´.(2)If q and r are algebraic integers of degree d of full rank and ℤ[qn] = ℤ[rn] for infinitely many n, then either q = ωr′ or q = Norm(r)2/dω/r′ , where r ′ is some conjugate of r and ω is some root of unity.(3)Let r be an algebraic integer of degree at most 3. Then there are at most 40 Pisot numbers q such that ℤ[q] = ℤ[r].(4)There are only finitely many Pisot-cyclotomic numbers of any fixed order.

On simultaneous rational approximations to a real number, its square, and its cube

We provide an upper bound on the uniform exponent of approximation to a triple (xi, xi^2, xi^3) by rational numbers with the same denominator, valid for any transcendental real number xi. This upper bound refines a previous result of Davenport and Schmidt. As a consequence, we get a sharper lower bound on the exponent of approximation of such a number xi by algebraic integers of degree at most 4.

Modular functions and the uniform distribution of CM points

(1) zd = { i √ d 2 if d ≡ 0 (mod 4), −1+i √ d 2 if d ≡ 3 (mod 4). The j-function has the remarkable property that j(zd) is an algebraic integer of degree h(−d), the class number of K = Q(zd). In fact, K(j(zd)) is the Hilbert class field of K [4]. The first few values of j(zd) are: j(z3) = 0, j(z4) = 12 , j(z7) = −153, j(z8) = 20, j(z11) = −323 j(z15) = −191025−85995√5 2 , j(z19) = −963, j(z20) = 632000+282880 √ 5,

On the absolute Mahler measure of polynomials having all zeros in a sector. III

Let $\alpha$ be an algebraic integer of degree $d$, not $0$ or a root of unity, all of whose conjugates $\alpha _i$ lie in a sector $\vert \arg z \vert \leq \theta$. In 1995, G. Rhin and C. Smyth computed the greatest lower bound $c(\theta )$ of the absolute Mahler measure ( $\prod _{i=1}^d \max (1, | \alpha _i |))^{1/d}$ of $\alpha$, for $\theta$ belonging to nine subintervals of $[0, 2 \pi /3]$. More recently, in 2004, G. Rhin and Q. Wu improved the result to thirteen subintervals of $[0, \pi ]$ and extended some existing subintervals. In this paper, for the first time we find a complete subinterval where $c(\theta )$ is known exactly, as well as a fourteenth subinterval. Moreover, we slightly extend further all the existing subintervals.

Approximation to real numbers by cubic algebraic integers I

In 1969, H. Davenport and W. M. Schmidt studied the problem of approximation to a real number ξ by algebraic integers of degree at most 3. They did so, using geometry of numbers, by resorting to the dual problem of finding simultaneous approximations to ξ and ξ2 by rational numbers with the same denominator. In this paper, we show that their measure of approximation for the dual problem is optimal and that it is realized for a countable set of real numbers ξ. We give several properties of these numbers including measures of approximation by rational numbers, by quadratic real numbers and by algebraic integers of degree at most 3. 2000 Mathematics Subject Classification 11J04 (primary), 11J13, 11J82 (secondary).

Simultaneous rational approximation to the successive powers of a real number

Let n be an integer ≥ 1 and let θ be a real number which is not an algebraic number of degree ≤ [n/2]. We show that there exist ϵ > 0 and arbitrary large real numbers X such that the system of linear inequalities |x0| ≤ X and |x0θj − xj| ≤ ϵX−1/[n/2] for 1 < j < n, has only the zero solution in rational integers x0,…, xn. This result refines a similar statement due to H. Davenport and W. M. Schmidt, where the upper integer part [n/2] is replaced everywhere by the integer part [n/2]. As a corollary, we improve slightly the exponent of approximation to 0 by algebraic integers of degree n + 1 over Q obtained by these authors.

Approximation simultanée d'un nombre et de son carré

In 1969, H. Davenport and W.M. Schmidt established a measure of the simultaneous approximation for a real number ξ and its square by rational numbers with the same denominator, assuming only that ξ is not rational nor quadratic over Q . Here, we show by an example, that this measure is optimal. We also indicate several properties of the numbers for which this measure is optimal, in particular with respect to approximation by algebraic integers of degree at most three. To cite this article: D. Roy, C. R. Acad. Sci. Paris, Ser. I 336 (2003).

Embedding dynamics for round-off errors near a periodic orbit.

We study the propagation of round-off errors near the periodic orbits of a linear map conjugate to a planar rotation with rational rotation number. We embed the two-dimensional discrete phase space (a lattice) in a higher-dimensional torus, where points sharing the same round-off error are uniformly distributed within finitely many convex polyhedra. The embedding dynamics is linear and discontinuous, with algebraic integer coefficients. This representation affords efficient algorithms for classifying and computing the orbits and their exact densities, which we apply to the case of rational rotation number with denominator 7, corresponding to certain algebraic integers of degree three. We provide evidence that the hierarchical arrangement of orbits previously detected in quadratic cases [Lowenstein et al., Chaos 7, 49-66 (1997)] disappears, and that the growth of the number of orbits with the period is algebraic.(c) 2000 American Institute of Physics.

On the Absolute Mahler Measure of Polynomials Having All Zeros in a Sector

Let α be an algebraic integer of degree d, not 0 or a root of unity, all of whose conjugates α i are confined to a sector |arg z| ≤ θ. We compute the greatest lower bound c(θ) of the absolute Mahler measure (Π i=1 d max(1, |α i |)) 1/d of α, for θ belonging to nine subintervals of [0, 2π/3]. In particular, we show that c(π/2) = 1.12933793, from which it follows that any integer α ¬= 1 and α ¬= e ±iπ/3 all of whose conjugates have positive real part has absolute Mahler measure at least c(π/2). This value is achieved for α satisfying α + 1/α = β 0 2 , where β 0 = 1.3247... is the smallest Pisot number (the real root of β 0 3 = β 0 + 1)

A Remark on a Paper by Wang: Another Surprising Property of 42

We give a counterexample to a bound quoted in Wang's paper on polynomial factorization over algebraic number fields. We also give an alternative to that bound which seems not to have been published before. Currently the best algorithms for factorizing polynomials over algebraic number fields perform a factorization over a finite field, refine this factorization by Hensel lifting to a suitable prime power, and finally deduce the true factorization. It is thus necessary to estimate how far the modular factorization has to be lifted so we can be sure of deducing the true irreducible factors. This estimate comes from a bound on the sizes of coefficients of the factors. We introduce some notation: f is the polynomial to be factorized over the field Q(0), where 0 is an algebraic integer of degree m with minimal polynomial mo (x) E Z[x]. Let 11011 = max{ 1q(0)| : 0: Q(0) -* C} (noting that the product of the q(0) is the trailing coefficient, so at least one 1X(0)I is at least one). If R = d=0 cjxj is a polynomial with complex coefficients, define jjRjj2 = Z ICj 2, following [2]. Recall Gauss's Lemma: let 0 be the ring of integers in Q(0); if f(x) E O[x] factorizes in Q(0)[x] then the factors can be taken to lie in O[x]. So we compute [1] the defect: the largest denominator, A, that occurs in 0, i.e., 0 C Z[0]. In fact, an integer multiple of A will suffice [4]. We now know how big the denominator can be, so we estimate the numerator. We can regard f(x) as an element of C[x] and determine the maximum magnitude, B, of any coefficient of any factor in C[x]. Thus our problem may be restated as: given that I Ejm=L ciu0/,Al < B for all embeddings Q(0) -* C, find a bound on Icil. Let M be the Vandermonde matrix z 1 0 02 ... 0m-1 1 02 02 0m-1 O1 m 02 m-1 so we can write Mc = b with b lying in a cube of side 2AB centered on 0. In [3] Wang quotes Weinberger (presumably [4]) as having proved (1) max K11 lIdet(M)I Received August 27, 1987; revised January 25, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 12D10. ?1988 American Mathematical Society 0025-5718/88 $1.00 + $.25 per page

On minimal diameters of algebraic integers in J-fields

Let α be an algebraic integer of degree n > 1 with conjugates α = α 1, α 2,…, α n . Let D( α) denote the diameter of the circumcircle of { α 1, α 2,…, α n } and diam( α) = max i, j | α i − α j |. Favard showed in 1930 that diam(α) ≥ √ 3 2 . Assuming that α belongs to a J-field, the author determines all such α for which D(α) < 4 cos ( π 9 ) − 1 . Hence, he finds all α for which diam(α) ≤ 2√3 cos ( π 9 ) − √ 3 2 , provided that α belongs to a J-field.

On a problem of Favard concerning algebraic integers

Let α be an algebraic integer of degree n &gt; 1 with conjugates α = α1, α2, …, αn. Let D(α) denote the diameter of {α1, …, αn} and diam . In 1929 Favard showed that diam and that, when n = 2 or 3, the minimal values of diam (α) are less than 2. The author shows that diam and He also finds all algebraic integers a with diam (α) ≤ 2 when n ≤ 5. Similar results are found for D(α) and, in particular, D(α) &gt; 2 when n = 5.

How to find algebraic integers whose conjugates lie near the unit circle?

In 1933 D. H. Lehmer [3], in connexion with a method for discovering large prime numbers, posed the following question. Let α be an algebraic integer of degree D with conjugates α = α 1 , α 2 , ..., α D and put[EQUATION]where a is the leading (positive) coefficient of the minimal polynomial of α.