Algebraic Number Fields and Rational Approximation

It is proved that the field of complex algebraic numbers has an isomorphic presentation computable in polynomial time. A similar fact is proved for the ordered field of real algebraic numbers. The constructed polynomially computable presentations are based on a natural presentation of algebraic numbers by rational polynomials. Also new algorithms for computing values of polynomials on algebraic numbers and for solving equations in one variable with algebraic coefficients are presented.

Matrix splitting with symmetry and dyadic framelet filter banks over algebraic number fields

The rational field Q is highly desired in many applications. Algorithms using the rational number field Q algebraic number fields use only integer arithmetics and are easy to implement. Therefore, studying and designing systems and expansions with coefficients in Q or algebraic number fields is particularly interesting. This paper discusses constructing quasi-tight framelets with symmetry over an algebraic field. Compared to tight framelets, quasi-tight framelets have very similar structures but much more flexibility in construction. Several recent papers have explored the structure of quasi-tight framelets. The construction of symmetric quasi-tight framelets directly applies the generalized spectral factorization of 2×2 matrices of Laurent polynomials with specific symmetry structures. We adequately formulate the latter problem and establish the necessary and sufficient conditions for such a factorization over a general subfield F of C, including algebraic number fields as particular cases. Our proofs of the main results are constructive and thus serve as a guideline for construction. We provide several examples to demonstrate our main results.

We prove that the field of complex algebraic numbers and the ordered field of real algebraic numbers have isomorphic presentations computable in polynomial time. For these presentations, new algorithms are found for evaluation of polynomials and solving equations of one unknown. It is proved that all best known presentations for these fields produce polynomially computable structures or quotient-structures such that there exists an isomorphism between them polynomially computable in both directions.

As is well-known, there are only finitely many isomorphic classes of finite subgroups in a given general linear group over the field of rational numbers. This result can be generalized to any algebraic number field. While the case of field of rational numbers is relatively well-studied, we still do not know much for general algebraic number field cases. In this article, we discuss the finiteness of isomorphic classes of finite subgroups of general linear groups over an algebraic number field. We give a method to calculate a multiplicative bound for the orders of finite subgroups and to classify finite cyclic subgroups.

In Part I, Mahler introduces his classification of complex numbers and the following two results are proved. Let $\vartheta\_1,\vartheta\_2,\ldots,\vartheta\_N$ be $N$ algebraic numbers that are linearly independent over the rationals and let $\lambda$ be a Liouville number. Then, the numbers $e^{\vartheta\_1},e^{\vartheta\_2},\ldots,e^{\vartheta\_N},\lambda$ are algebraically independent over the field of algebraic numbers. Let $z$ be the real logarithm of a positive rational number not equal to one and let $\lambda$ be a Liouville number. Then, $z$ and $\lambda$ are algebraically independent over the field of algebraic numbers. Part II continues the study of the same title by giving various bounds on polynomials evaluated at logarithms and exponentials. A new proof of the transcendence of $\pi$ is given as an application. Reprint of the author's papers \[J. Reine Angew. Math. 166, 118--136 (1931; Zbl 0003.15101; JFM 57.0242.03); ibid. 166, 137--150 (1932; Zbl 0003.38805; JFM 58.0207.01)].

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.

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

We present algorithms that compute all irreducible factors of degree ≤ d of supersparse (lacunary) multivariate polynomials in n variables over an algebraic number field in deterministic polynomial-time in (l+d)n, where l is the size of the input polynomial. In supersparse polynomials, the term degrees enter logarithmically as their numbers of binary digits into the size measure l. The factors are again represented as supersparse polynomials. If the factors are represented as straight-line programs or black box polynomials, we can achieve randomized polynomial-time in (l+d)O(1). Our approach follows that by H. W. Lenstra, Jr., on computing factors of univariate supersparse polynomials over algebraic number fields. We generalize our ISSAC 2005 results for computing linear factors of supersparse bivariate polynomials over the rational numbers by appealing to recent lower bounds on the height of algebraic numbers and to a special case of the former Lang conjecture.

On the modular forms of weight 1/2 over algebraic number fields

In this work, the values of certain lacunar power series with rational coefficients for 𝑈𝒎-number arguments were determined to be either in a particular algebraic number field or in the set of transcendental numbers under specific circumstances in the complex numbers field. The result was also applied on some of the lacunary power series with coefficients in an algebraic number field. Roth's theorem which is the essential result in Diophantine approximation to algebraic numbers was used to reach the present results.

In this paper, we prove a weak form of the conjecture generalised to algebraic number fields. Given integers satisfying , Stewart and Yu were able to give an exponential bound in terms of the radical over the integers (Stewart and Yu [Math. Ann. 291 (1991), 225–230], Stewart and Yu [Duke Math. J. 108 (2001), no. 1, 169–181]), whereas Győry was able to give an exponential bound in the algebraic number field case for the projective height in terms of the radical for algebraic numbers (Győry [Acta Arith. 133 (2008), 281–295]). We generalise Stewart and Yu's method to give an improvement on Győry's bound for algebraic integers over the Hilbert Class Field of the initial number field K. Given algebraic integers in a number field K satisfying , we give an upper bound for the logarithm of the projective height in terms of norms of prime ideals dividing , where L is the Hilbert Class Field of K. In many cases, this allows us to give a bound in terms of the modified radical as given by Masser (Proc. Amer. Math. Soc. 130 (2002), no. 11, 3141–3150). Furthermore, by employing a recent bound of Győry (Publ. Math. Debrecen 94 (2019), 507–526) on the solutions of S‐unit equations, our estimates imply the upper bound where is an effectively computable constant. Further, given conditions on the largest prime ideal dividing , we obtain a sub‐exponential bound for in terms of the radical. Independently, as a direct application of his bounds on the solutions of S‐unit equations(Győry ([Publ. Math. Debrecen 94 (2019), 507–526]), Győry (Publ. Math. Debrecen 100 (2022), 499–511) also attains results mentioned above, including the above inequality, but over the base field K, as discussed in Section 6. As a consequence of our results, we will give an application to the effective Skolem–Mahler–Lech problem and give an improvement to a result by Lagarias and Soundararajan (J. Théor. Nombres Bordeaux 23 (2011), no. 1, 209–234) on the XYZ conjecture.

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.

This paper provides a generalization of the Hua Loo-Keng estimation method of rational trigonometric sums with a polynomial in exponent in algebraic number fields, which are extensions of the field of rational numbers. In the ring of integers of this algebraic number field we consider integer and fractional ideals. For a complete system of residues for any integer ideal, Hua Loo-Keng proved an analogue of the Euler–Fourier formula, which, using resultsregarding the multiplicity of roots of a polynomial congruence modulo a prime ideal (“Hua Loo-Keng trees”), allows the problem to be reduced to the p-adic lifting of solutions, and this allows us to reduce the problem of estimating the sum to estimating the number of solutions of polynomial congruences modulo a power of a prime ideal. Furthermore, building upon Chen Jingrun’s estimates in the field of rational numbers, we obtain improved constants for similar estimates in algebraic numeric fields.

A fundamental problem of experimental algebraic number theory is that of determining the unitand class group of an algebraic number field K. To solve this problem for large classes of number fields, effective algorithms are needed. Such algorithms mostly rest on methods from geometry of numbers, as this is the case with the method of Pohst and Zassenhaus. They require knowledge of an integral basis for K/~. Such bases of number fields canbe constructed in an efficient manner by applying an algorithm of Ford and Zassenhaus° Here local considerations play an important role. That is why one has to perform calculations not only over the completion of Q with respect to ordinary absolute value, i.e. over the real field IR, but at the same time also over the completions of Q with respect to the p-adic valuations, i.e. over the p-adic fields Up. Furthermore, a p-adic analogue of the complex number field ~ the latter being the ~]g~h~ closure of ~rR, is the completion Cp of the algebraic closure ~p of ~p. In this connection we recall that Hasse, in his fundamental book on algebraic number theory, uses a simultaneous treatment of the completions IR and @p as a foundation of the arithmetic in algebraic number fields.