On the solution of units and index form equations in algebraic number fields. (Über das Lösen von Einheiten- und Indexformgleichungen in algebraischen Zahlkörpern.) (German) Zbl 0952.11032

Wildanger, K.

J. Number Theory 82, No. 2, 188-224 (2000).

Many classical diophantine equations can be reduced to unit equations of type \[ \alpha a +\beta b =1 \] where \(\alpha,\beta\) are non-zero elements of an algebraic number field \(K\), and the variables are units of \(K\). These units can be written as a power product of the fundamental units of \(K\): \[ a=\eta_1^{a_1}\ldots \eta_r^{a_r}, b=\eta_1^{b_1}\ldots \eta_r^{b_r} \] Using Baker’s method one can derive an upper bound for the absolute values of the exponents \(a_j,b_j\). This huge upper bound (about \(10^{20}\) even in the simplest cases) can be reduced by the LLL reduction algorithm to a bound of magnitude \(10^2\) up to \(10^4\). For small unit ranks this is sufficient to find all solutions. For higher \(r\) it is still a hard problem to test all possible values of the exponents under the reduced bound. For \(r\leq 4\) one can use sieve methods.
For about \(r>5\) it was formerly almost impossible to overcome the difficulty of testing the small solutions. Using the author’s algorithm described in the paper it is now feasible to solve unit equations up to unit rank about 10. Therefore the paper yields a breakthrough in the area “constructive solution of diophantine equations”, making possible to solve much more complicated equations as before. The method is already known as “Wildanger’s method” having a couple of applications.
The essence of the method is that the author shows that the possible solutions are contained in certain thin ellipsoids that are easy to enumerate. Here an algorithm of U. Fincke and M. Pohst [Math. Comput. 44, 463-471 (1985; Zbl 0556.10022)] also plays an important role.

Reviewer: I.Gaál (Debrecen)

MSC:

11Y50	Computer solution of Diophantine equations
11D57	Multiplicative and norm form equations

Keywords:

unit equation; index form equation; ellipsoid method; Baker’s method; LLL reduction algorithm

Citations:

Zbl 0556.10022

Software:

KANT/KASH

References:

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