Solving superelliptic Diophantine equations by Baker's method

We describe a method for complete solution of the superelliptic Diophantine equation ay^p=f(x). The method is based on Baker's theory of linear forms in the logarithms. The characteristic feature of our approach (as compared with the classical method) is that we reduce the equation directly to the linear forms in logarithms, without intermediate use of Thue and linear unit equations. We show that the reduction method of Baker and Davenport [3] is applicable for superelliptic equations, and develop a very efficient method for enumerating the solutions below the reduced bound. The method requires computing the algebraic data in number fields of degree pn(n-1)/2 at most; in many cases this number can be reduced. Two examples with p=3 and n=4 are given.     

代数数的有理逼近及其在求解丢番图方程中的应用

在本文中,我们利用Ｔｈｕｅ－Ｓｉｅｐｅｌ方法研究一类代数数的有理逼近,证明了对此代数数的有效有一逼近,最后我们利用此结果研究了ｄｉｏｐｈａｎｔｉｎｅ方程。ａｘ＾２－ｂｙ＾４＝－１得出关于此方程完整的结论。     

On the resolution of relative Thue equations

An efficient algorithm is given for the resolution of relative Thue equations. The essential improvement is the application of an appropriate version of Wildanger's enumeration procedure based on the ellipsoid method of Fincke and Pohst.

Recently relative Thue equations have gained an important application, e.g., in computing power integral bases in algebraic number fields. The presented methods can surely be used to speed up those algorithms.

The method is illustrated by numerical examples.

     

Solving Thue equations without the full unit group

The main problem when solving a Thue equation is the computation of the unit group of a certain number field. In this paper we show that the knowledge of a subgroup of finite index is actually sufficient. Two examples linked with the primitive divisor problem for Lucas and Lehmer sequences are given.

     

Diophantine equations over global function fields I: The Thue equation

We solve completely Thue equations in function fields over arbitrary finite fields. In the function field case such equations were formerly only solved over algebraically closed fields (of characteristic zero and positive characteristic). Our method can be applied to similar types of Diophantine equations, as well.     

六次含参Thue方程的解(英文)

本文研究了含参的六次Thue方程.利用初等方法和简单的代数数有理逼近方法彻底求解了该方程,从而推广了Alan Togb′e的结果.     

On a Family of Thue Equations Over Function Fields

Many families of parametrized Thue equations over number fields have been solved recently. In this paper we consider for the first time a family of Thue equations over a polynomial ring. In particular, we calculate all solutions of X(X-Y)(X-(T+x)Y)+Y³=1+xT(1-T)X(X-Y)(X-(T+\xi)Y)+Y^3=1+\xi T(1-T) over \Bbb C[T]{\Bbb C}[T] for all x ? \Bbb C\xi\in{\Bbb C} .     

On the solutions of a parametric family of cubic Thue equations

In this paper, we solve a family of Diophantine equations associated with families of number fields of degree 3. In fact, we use Baker’s method find all solutions to the Thue equation

On a Family of Thue Equations Over Function Fields

Many families of parametrized Thue equations over number fields have been solved recently. In this paper we consider for the first time a family of Thue equations over a polynomial ring. In particular, we calculate all solutions of over for all . The first author was supported by the Austrian Science Foundation, grants S8307-MAT and J2407-N12. The second author was supported by the Austrian Science Foundation, grant S8307-MAT.     

Thue inequalities with a small number of primitive solutions

Several upper bounds are known for the numbers of primitive solutions (x; y) of the Thue equation (1) j F(x; y) j = m and the more general Thue inequality (3) 0 < j F(x; y) j m. A usual way to derive such an upper bound is to make a distinction between "small" and "large" solutions, according as max( j x j ; j y j ) is smaller or larger than an appropriate explicit constant Y depending on F and m; see e.g. [1], [11], [6] and [2]. As an improvement and generalization of some earlier results we give in Section 1 an upper bound of the form cn for the number of primitive solutions (x; y) of (3) with max( j x j ; j y j )Y0 , wherec 25 is a constant and n denotes the degree of the binary form F involved (cf. Theorem 1). It is important for applications that our lower bound Y0 for the large solutions is much smaller than those in [1], [11], [6] and [4], and is already close to the best possible in terms of m. ByusingTheorem1 we establish in Section 2 similar upper bounds for the total number of primitive solutions of (3), provided that the height or discriminant of F is suficiently large with respect to m (cf. Theorem 2 and its corollaries). These results assert in a quantitative form that, in a certain sense, almost all inequalities of the form (3) have only few primitive solutions. Theorem 2 and its consequences are considerable improvements of the results obtained in this direction in [3], [6], [13] and [4]. The proofs of Theorems 1 and 2 are given in Section 3. In the proofs we use among other things appropriate modifications and refenements of some arguments of [1] and [6].     

Class Numbers of Orders in Cubic Fields

In this paper it is shown that the sum of class numbers of orders in complex cubic fields obeys an asymptotic law similar to the prime numbers as the bound on the regulators tends to infinity. Here only orders are considered which are maximal at two given primes. This result extends work of Sarnak in the real quadratic case. It seems to be the first asymptotic result on class numbers for number fields of degree higher than two.     

Simple families of Thue inequalities

We use the hypergeometric method to solve three families of Thue inequalities of degree 3, 4 and 6, respectively, each of which is parametrized by an integral parameter. We obtain bounds for the solutions, which are astonishingly small compared to similar results which use estimates of linear forms in logarithms.

     

A parametric family of cubic Thue equations

In this paper, we solve a family of Diophantine equations associated with families of number fields of degree 3. In fact, we find all solutions to the Thue equation

     

Axiomatisations simples des theories des corps de rolle

Using Becker's results we obtain here a simple first order axiomatization, looking like those by Artin-Schreier and also written in the language of fields, for the theory of Rolle fields (i.e. fields with the Rolle's property for every order). In fields having a finite number of orders, we characterize Rolle fields as those which are pythagorean at level 4 and do not admit any algebraic extension of odd degree. Then we give an axiomatization for Rolle fields having exactly 2ⁿ orders (n≥0); in fact, for n=0 we recover an axiomatization of the theory of real-closed fields and for n=1 we get exactly an axiomatization given for the theory of chain-closed fields by the author in [G1]. Finally we prove that a Rolle field with exactly 2ⁿ orders is the intersection of n+1 real closures of the field.      

Nonrepetitive vertex colorings of graphs

We prove new upper bounds on the Thue chromatic number of an arbitrary graph and on the facial Thue chromatic number of a plane graph in terms of its maximum degree.     

On orders in number fields: Picard groups,ring class fields and applications

We focus on orders in arbitrary number fields, consider their Picard groups and finally obtain ring class fields corresponding to them. The Galois group of the ring class field is isomorphic to the Picard group.As an application, we give criteria of the integral solvability of the diophantine equation p = x2+ ny2 over a class of imaginary quadratic fields where p is a prime element.     

Simplifying method for algebraic approximation of certain algebraic numbers

A new simple method for approximating certain algebraic numbers is developed. By applying this method, an effective upper bound is derived for the integral solutions of the quartic Thue equation with two parameters $tx^4 - 4sx^3 y - 6tx^2 y^2 + 4sxy^3 + ty^4 = N$ , where s > 32t ³. As an application, Ljunggren’s equation is solved in an elementary way.     

Recurrent Methods for Constructing Irreducible Polynomials over GF(2s)

The paper is devoted to some results concerning the constructive theory of the synthesis of irreducible polynomials over Galois fields GF(q), q=2^s. New methods for the construction of irreducible polynomials of higher degree over GF(q) from a given one are worked out. The complexity of calculations does not exceed O(n³) single operations, where n denotes the degree of the given irreducible polynomial. Furthermore, a recurrent method for constructing irreducible (including self-reciprocal) polynomials over finite fields of even characteristic is proposed.     

Representation of algebraic integers by ternary quadratic forms

The discrete ergodic method is generalized to totally positive-ternary quadratic forms over totally real algebraic number fields. We obtain estimates for the number of representations of elements in maximal orders of such number fields which are precise in the sense of the order of growth. We prove that the representations are asymptotically uniformly distributed with respect to a given module.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 121, pp. 157–168, 1983.     

On relations between zeros of Galois polynomials

A conjecture ofH. Kleiman says that over certain fields a Galois equation of degree 3 is uniquely determined by its root polynomials. We prove this conjecture for prime degrees 3 and a somewhat smaller class of fields than Kleiman's. In this situation, the ideal of all relations between zeros of the equation has a basis containing root polynomials only, not the equation itself. Giving a large class of counterexamples of degree 4, we disprove Kleiman's conjecture in general.