Algebraic Numbers Of Degree Research Articles

Invariant set generated by a nonreal number is everywhere dense

A set of complex numbers $S$ is called invariant if it is closed under addition and multiplication, namely, for any $x, y \in S$ we have $x+y \in S$ and $xy \in S$ . For each $s \in {\mathbb {C}}$ the smallest invariant set ${\mathbb {N}}[s]$ containing $s$ consists of all possible sums $\sum _{i \in I} a_i s^i$ , where $I$ runs over all finite nonempty subsets of the set of positive integers ${\mathbb {N}}$ and $a_i \in {\mathbb {N}}$ for each $i \in I$ . In this paper, we prove that for $s \in {\mathbb {C}}$ the set ${\mathbb {N}}[s]$ is everywhere dense in ${\mathbb {C}}$ if and only if $s \notin {\mathbb {R}}$ and $s$ is not a quadratic algebraic integer. More precisely, we show that if $s \in {\mathbb {C}} \setminus {\mathbb {R}}$ is a transcendental number, then there is a positive integer $n$ such that the sumset ${\mathbb {N}} t^n+{\mathbb {N}} t^{2n} +{\mathbb {N}} t^{3n}$ is everywhere dense in ${\mathbb {C}}$ for either $t=s$ or $t=s+s^2$ . Similarly, if $s \in {\mathbb {C}} \setminus {\mathbb {R}}$ is an algebraic number of degree $d \ne 2, 4$ , then there are positive integers $n, m$ such that the sumset ${\mathbb {N}} t^n+{\mathbb {N}} t^{2n} +{\mathbb {N}} t^{3n}$ is everywhere dense in ${\mathbb {C}}$ for $t=ms+s^2$ . For quadratic and some special quartic algebraic numbers $s$ it is shown that a similar sumset of three sets cannot be dense. In each of these two cases the density of ${\mathbb {N}}[s]$ in ${\mathbb {C}}$ is established by a different method: for those special quartic numbers, it is possible to take a sumset of four sets.

Absorbing games with irrational values

Many classes of two-player zero-sum stochastic games have the orderfield property; that is, if all payoffs and transitions belong to some field, so does the limit value. Is it also the case for absorbing games? No: In this note, we exhibit m×m absorbing games with rational data whose limit values are algebraic of degree m, for each m∈N⁎. Furthermore, we provide maximal conditions for the orderfield property to hold, namely if transitions are deterministic and one player has at most two actions. Last, we prove that any algebraic number of degree 2 is the limit value of a 2×2 absorbing game, which leads to the conjecture that any algebraic number of degree m is the limit value of an m×m absorbing game.

Stewart’s theorem revisited: suppressing the norm $$\pm 1$$ hypothesis

Let \(\gamma\) be an algebraic number of degree 2 and not a root of unity. In this note we show that there exists a prime ideal \({\mathfrak {p}}\) of \({\mathbb {Q}}(\gamma )\) satisfying \(\nu _{\mathfrak {p}}(\gamma ^n-1)\ge 1\), such that the rational prime p underlying \({\mathfrak {p}}\) grows quicker than n.

Exponents of Diophantine approximation in dimension 2 for a general class of numbers

We study the Diophantine properties of a new class of transcendental real numbers which contains, among others, Roy's extremal numbers, Bugeaud-Laurent Sturmian continued fractions, and more generally the class of Sturmian type numbers. We compute, for each real number $\xi$ of this set, several exponents of Diophantine approximation to the pair $(\xi,\xi^2)$, together with $\omega_2^*(\xi)$ and $\widehat{\omega}_2^*(\xi)$, the so-called ordinary and uniform exponent of approximation to $\xi$ by algebraic numbers of degree $\leq 2$. As an application, we get new information on the set of values taken by $\widehat{\omega}_2^*$ at transcendental numbers, and we give a partial answer to a question of Fischler about his exponent $\beta_0$.

On constructing Greek ladders to approximate any real algebraic number

ABSTRACT In 2005, Osler et al. showed that algebraic irrational numbers of the form for can be approximated using specific Greek ladders (Osler, T. J., Wright, M., & Orchard, M. (2005). Theon's ladder for any root. International Journal of Mathematical Education in Science and Technology, 36(4), 389–398. https://doi.org/10.1080/00207390512331325969). In 2018, Herzinger et al., showed that every algebraic number of degree 2 or less can be approximated using rung ratios generated by a specific Greek ladder (Herzinger, K., Kunselman, C., & Pierce, I. (2018). Greek ladders via linear algebra. International Journal of Mathematical Education in Science and Technology, 49(7), 1119–1132. https://doi.org/10.1080/0020739X.2018.1440326). Using techniques from linear algebra, in particular the companion matrix of a polynomial, we will prove that every real algebraic number can be approximated by a Greek ladder, as well as show how to construct such a ladder.

On the frequency of height values

We count algebraic numbers of fixed degree d and fixed (absolute multiplicative Weil) height {mathcal {H}} with precisely k conjugates that lie inside the open unit disk. We also count the number of values up to {mathcal {H}} that the height assumes on algebraic numbers of degree d with precisely k conjugates that lie inside the open unit disk. For both counts, we do not obtain an asymptotic, but only a rough order of growth, which arises from an asymptotic for the logarithm of the counting function; for the first count, even this rough order of growth exists only if k in {0,d} or gcd (k,d) = 1. We therefore study the behaviour in the case where 0< k < d and gcd (k,d) > 1 in more detail. We also count integer polynomials of fixed degree and fixed Mahler measure with a fixed number of complex zeroes inside the open unit disk (counted with multiplicities) and study the dynamical behaviour of the height function.

An improved bound in Wirsing’s problem

We improve the lower bound for the classical exponent of approximation w n ∗ w_{n}^{\ast } connected to Wirsing’s famous problem on approximation to real numbers by algebraic numbers of degree at most n n . Our bound exceeds n / 3 ≈ 0.5773 n n/\sqrt {3}\approx 0.5773n and thus provides a considerable qualitative improvement to previous bounds of order n / 2 + O ( 1 ) n/2+O(1) . We further establish new relations between several classical exponents of approximation.

Lower bounds for the number of vectors with algebraic coordinates near smooth surfaces

Let z = f ( x , y ) be a surface in three-dimensional Euclidean space. Consider a neighborhood V of this surface, whose points satisfy the inequality | f ( x , y ) - z| < Q -Y , where 0 < у < 1 and Q is a sufficiently large positive integer. In the works of Huxley, Beresnevich, Velani, the distribution of rational points in V has been started. In this article, we study the distribution of points with real conjugate algebraic coordinates ᾱ = α 1 α 2 α 3 in V. For some c 1 = c 1 ( n ), a lower bound is obtained in the form of c 2 Q n+1-Y for the number of algebraic numbers of degree n ≥ 3 and of height at most c 3 Q .

A note on two linear forms

We prove a result on approximations to a real number $\theta$ by algebraic numbers of degree $\le 2$ in the case when we have information about the uniform Diophantine exponent $\hat{\omega}$ for the linear form $x_0 +\theta x_1+\theta^2x_2$.

On relations for rings generated by algebraic numbers and their conjugates

Let \(\alpha \) be an algebraic number of degree \(d\) with minimal polynomial \(F \in \mathbb {Z}[X]\), and let \(\mathbb {Z}[\alpha ]\) be the ring generated by \(\alpha \) over \(\mathbb {Z}\). We are interested whether a given number \(\beta \in \mathbb {Q}(\alpha )\) belongs to the ring \(\mathbb {Z}[\alpha ]\) or not. We give a practical computational algorithm to answer this question. Furthermore, we prove that a rational number \(r/t \in \mathbb {Q}\), where \(r \in \mathbb {Z}, t \in \mathbb {N}, \gcd (r, t) = 1\), belongs to the ring \(\mathbb {Z}[\alpha ]\) if and only if the square-free part of its denominator \(t\) divides all the coefficients of the minimal polynomial \(F \in \mathbb {Z}[X]\) except for the constant coefficient \(F(0)\) that must be relatively prime to \(t\), namely \(\gcd (F(0),t)=1\). We also study the question when the equality \(\mathbb {Z}[\alpha ] = \mathbb {Z}[\alpha ']\) for algebraic numbers \(\alpha , \alpha '\) conjugates over \(\mathbb {Q}\) holds. In particular, it is shown that for each \(d \in \mathbb {N}\), there are conjugate algebraic numbers \(\alpha , \alpha '\) of degree \(d\) satisfying \(\mathbb {Q}(\alpha ) = \mathbb {Q}(\alpha ')\) and \(\mathbb {Z}[\alpha ] \ne \mathbb {Z}[\alpha ']\). The question concerning the equality \(\mathbb {Z}[\alpha ]=\mathbb {Z}[\alpha ']\) is answered completely for conjugate quadratic pairs \(\alpha ,\alpha '\) and also for conjugate pairs \(\alpha , \alpha '\) of cubic algebraic integers.

Automatic continued fractions are transcendental or quadratic

We establish new combinatorial transcendence criteria for continued fraction expansions. Let $\alpha = [0; a_1, a_2,...]$ be an algebraic number of degree at least three. One of our criteria implies that the sequence of partial quotients $(a_{\ell})_{\ell \ge 1}$ of $\alpha$ cannot be generated by a finite automaton, and that the complexity function of $(a_{\ell})_{\ell \ge 1}$ cannot increase too slowly.

COMPLEX NUMBERS WITH BOUNDED PARTIAL QUOTIENTS

AbstractConjecturally, the only real algebraic numbers with bounded partial quotients in their regular continued fraction expansion are rationals and quadratic irrationals. We show that the corresponding statement is not true for complex algebraic numbers in a very strong sense, by constructing, for every even degree $d$, algebraic numbers of degree $d$ that have bounded complex partial quotients in their Hurwitz continued fraction expansion. The Hurwitz expansion is the complex generalization of the nearest integer continued fraction for real numbers. In the case of real numbers the boundedness of regular and nearest integer partial quotients is equivalent.

A degree problem for two algebraic numbers and their sum

For all but one positive integer triplet (a; b; c) with a < b < c and b < 6, we decide whether there are algebraic numbers α,β and γ of degrees a, b and y, respectively, such that α+β+γ = 0. The undecided case (6; 6; 8) will be included in another paper. These results imply, for example, that the sum of two algebraic numbers of degree 6 can be of degree 15 but cannot be of degree 10. We also show that if a positive integer triplet (a; b; c) satisfies a certain triangle-like inequality with respect to every prime number then there exist algebraic numbers α,β γ of degrees a, b, c such that α+β+γ = 0. We also solve a similar problem for all (a; b; c) with a < b < c and b <6 by finding for which a, b, c there exist number fields of degrees a and b such that their compositum has degree c. Further, we have some results on the multiplicative version of the first problem, asking for which triplets (a; b; c) there are algebraic numbers and α, β and γ of degrees a, b and c, respectively, such that αβγ = 1.

COUNTING MULTISECTIONS IN CONIC BUNDLES OVER A CURVE DEFINED OVER 𝔽q

For a given conic bundle X over a curve C defined over 𝔽q, we count irreducible branch covers of C in X of degree d and height e ≫ 1. As a special case, we get the number of algebraic numbers of degree d and height e over the function field 𝔽q(C).

On certain infinite extensions of the rationals with Northcott property

A set of algebraic numbers has the Northcott property if each of its subsets of bounded Weil height is finite. Northcott's Theorem, which has many Diophantine applications, states that sets of bounded degree have the Northcott property. Bombieri, Dvornicich and Zannier raised the problem of finding fields of infinite degree with this property. Bombieri and Zannier have shown that $\IQ_{ab}^{(d)}$, the maximal abelian subfield of the field generated by all algebraic numbers of degree at most $d$, is such a field. In this note we give a simple criterion for the Northcott property and, as an application, we deduce several new examples, e.g. $\IQ(2^{1/d_1},3^{1/d_2},5^{1/d_3},7^{1/d_4},11^{1/d_5},...)$ has the Northcott property if and only if $2^{1/d_1},3^{1/d_2},5^{1/d_3},7^{1/d_4},11^{1/d_5},...$ tends to infinity.

D. K. Faddeev and St. Petersburg algebraic school

A review of scientific achievements of D.K. Faddeev. DOI: 10.3103/S1063454108010019 TO THE 100th ANNIVERSARY OF BIRTHDAY OF DMITRII KONSTANTINOVICH FADDEEV On June 30, 2007, we celebrated the 100th anniversary of birthday of an outstanding mathematician, Corresponding Member of the USSR Academy of Sciences, Dmitrii Konstantinovich Faddeev (1907– 1989). Dmitrii Kostantinovich is the founder of the Leningrad–St. Petersburg algebraic school, which has gained world-wide recognition. Disciples of that school obtained a lot of scientific results of highest quality. He was born into the family of a St. Petersburg engineer in a small town of Yukhnov in Smolensk region (now Kaluga region). At the age of sixteen, he entered the mathematical department of Petrograd university and since that time to the end of his life (with a short interval in the late 1920s) his multifarious creative activity was closely related to our university, of which he became a professor in 1933. For a long period of time, Dmitrii Konstantinovich headed the chair of Higher algebra and number theory; during several years, he was dean of the department of mathematics and mechanics. In 1932, he started working at the Steklov mathematical institute of the USSR Academy of Sciences, which was at that time known as the Steklov physico-mathematical institute. From the moment of foundation of the Leningrad branch of the institute (LOMI, now POMI RAS) in 1940 to the end of his life, D.K. Faddeev was one of the most prominent researches there. For many years he headed the laboratory of algebraic methods, which became a generally recognized center of algebraic studies. The area of scientific interests of D.K. Faddeev was extremely wide. First of all, we remember him as one of the most prominent algebraists of that time. His investigations covered almost all subfields of algebra. His first results relate to the theory of Diophantine equations, which became a subject of his study under the influence of his scientific advisor B.N. Delone. These results are really impressive: he has essentially extended the class of equations of degree three and four which can be completely solved. As is known, solutions to binary equations of degree three form a group under the natural multiplication, and the Mordell– Weil classical theorem states that this group is finitely generated. However, the rank of this group is hard to calculate. In a number of cases, estimates for this rank obtained by D.K. Faddeev allow one to find all solutions to the equation. For instance, equations x 3 + y 3 = A are completely solved for all A ≤ 50. To illustrate the significance of this result, we note that, before Faddeev’s works, it was known only that, for some values of A , this equation possesses only trivial solutions. There is another elegant result concerning the equation x 4 + Ay 4 = ± 1: D.K. Faddeed demonstrated that, for any integer A , this equation has at most one nontrivial solution; moreover, this solution corresponds to the principal unit of a certain purely imaginary field of algebraic numbers of degree four and exists only when the principal unit is trinomial. These results were published in the remarkable monograph “The Theory of Irrationalities of the Third Degree” written by Dmitrii Konstantinovich in co-authorship with B.N. Delone. In later years, D.K. Faddeev continued investigating Diophantine equations. In a series of works he proves that the group of classes of divisors of degree zero for the Fermat curve of degree 4, 5, and 7 is finite. This yields very powerful results concerning the equations themselves. For instance, there exists only a finite number of quadratic fields where equation x 4 + y 4 = 1 possesses a nontrivial solution and for each of these fields, the number of solutions is finite. Results of this kind had never been obtained before. Of course, by now, many classical problems of the theory of Diophantine equations have been solved and in our days these results are viewed from another standpoint, but at that time, they were really outstanding. Another sphere of interests of D.K. Faddeev in the early years of his scientific activity was Galois theory. He was particularly interested in the so-called inverse problem of Galois theory (which still remains not completely solved). This problem consists in constructing an extension of a given base field with a pre-

Infinitely Many Periodic Solutions of Nonlinear Equations of Suspended Strings

We shall deal with a nonlinear equation of a suspended string with a special density to which a nonlinear time-independent external force is applied. We shall show the existence of infinitely many time-periodic solutions, near time-periodic solutions which are independent of position. The set of the periods is constructed by using algebraic numbers of degree 2 and the continued fractions.

On approximation of real numbers by algebraic numbers of bounded degree

Dirichlet proved that for any real irrational number ξ there exist infinitely many rational numbers p / q such that | ξ − p / q | < q −2 . The correct generalization to the case of approximation by algebraic numbers of degree ⩽ n, n > 2 , is still unknown. Here we prove a result which improves all previous estimates concerning this problem for n > 2 .

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 d'un nombre réel par des unités algébriques

In this paper, we prove that for any real number ξ, which is not an algebraic number of degree , there exist infinitely many real algebraic units α of degree n + 1 such that . We also show how the flexibility of H. Davenport and W. M. Schmidt’s method allows to replace, with the same exponent of approximation, units of degree over Z (i.e. elements α with both α and integral over Z) by units of degree over a finite intersection .