## 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

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

Zbl 0556.10022

KANT/KASH
Full Text:

### References:

 [1] Baker, A.; Wüstholz, G., Logarithmic forms and group varieties, J. Reine Angew. Math., 442, 19-62 (1993) · Zbl 0788.11026 [2] Bremner, A., On power bases in cyclotomic number fields, J. Number Theory, 28, 288-298 (1988) · Zbl 0637.12001 [3] Daberkow, M.; Fieker, C.; Klüners, J.; Pohst, M.; Roegener, K.; Wildanger, K., KANT V4, J. Symbolic Comput., 24, 267-283 (1997) · Zbl 0886.11070 [4] Evertse, J. H., Upper bounds for the number of solutions of Diophantine equations, CWI Tract (1983), Stichting Mathematisch Centrum: Stichting Mathematisch Centrum Amsterdam · Zbl 0517.10016 [5] Fincke, U.; Pohst, M., Improved methods for calculating vectors of short length in a lattice, including a complexity analysis, Math. Comp., 44, 463-471 (1985) · Zbl 0556.10022 [6] Gaál, I., Computing all power integral bases in orders of totally real cyclic sextic number fields, Math. Comp, 65, 801-822 (1996) · Zbl 0857.11069 [7] Gaál, I., Comuting elements of given index in totally complex cyclic sextic fields, J. Symbolic Comput., 20, 61-69 (1995) · Zbl 0857.11068 [8] Gaál, I.; Pethő, A.; Pohst, M., On the resolution of index form equations in quartic number fields, J. Symbolic Comput., 16, 563-584 (1993) · Zbl 0808.11023 [9] Gaál, I.; Pethő, A.; Pohst, M., Simultaneous representation of integers by a pair of ternary quadratic forms—With an application to index form equations in quartic number fields, J. Number Theory, 57, 90-104 (1996) · Zbl 0853.11023 [10] Gaál, I.; Pohst, M., On the resolution of index form equations in sextic fields with an imaginary subfield, J. Symbolic Comput., 22, 425-434 (1996) · Zbl 0873.11025 [11] Gaál, I.; Schulte, N., Computing all power integral bases of cubic fields, Math. Comp., 53, 689-696 (1989) · Zbl 0677.10013 [12] Gras, M. N., Non monogénéité de l’anneau des entiers des extensions cycliques de $$Q$$ de degré premier $$l$$⩾5, J. Number Theory, 23, 347-353 (1986) [13] Győry, K., Sur l’irréductibilité d’une classe des polynômes, I, Publ. Math. Debrecen, 18, 289-307 (1971) · Zbl 0251.12104 [14] Győry, K., Sur les polynômes à coefficients entiers et de discriminant donné, II, Publ. Math. Debrecen, 21, 125-144 (1974) · Zbl 0303.12001 [15] Lenstra, A. K.; Lenstra, H. W.; Lovász, L., Factoring polynomials with rational coefficients, Math. Ann., 261, 515-534 (1982) · Zbl 0488.12001 [16] Nagell, T., Sur une propriété des unités d’un corps algébrique, Ark. Mat., 5, 343-356 (1964) · Zbl 0128.03403 [17] Nagell, T., Sur les unités dans les corps biquadratiques primitifs du premier rang, Ark. Mat., 7, 359-394 (1968) · Zbl 0164.35201 [18] Nagell, T., Sur un type particulier d’unités algébriques, Ark. Mat., 8, 163-184 (1969) · Zbl 0213.06901 [19] Niklasch, G., Einheitengleichungen in kommutativen Ringen (1991), Technische Universität München: Technische Universität München München [20] G. Niklasch, Family portraits of exceptional units, Manuskript.; G. Niklasch, Family portraits of exceptional units, Manuskript. · Zbl 0899.11053 [21] Pohst, M.; Zassenhaus, H., Algorithmic Algebraic Number Theory (1989), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0685.12001 [22] Siegel, C. L., Über einige Anwendungen diophantischer Approximationen, Abh. Akad. Wiss. Phys.-Math., 1, 209-266 (1929) [23] Smart, N. P., The solution of triangularly connected decomposable form equations, Math. Comp., 64, 819-840 (1995) · Zbl 0831.11027 [24] Smart, N. P., Discriminant form equations in number fields of degree greater than four, J. Symbolic Comput., 21, 367-374 (1996) · Zbl 0867.11016 [25] Sprindžuk, V. G., Classical Diophantine Equations. Classical Diophantine Equations, Lecture Notes in Mathematics, 1559 (1990), Springer-Verlag: Springer-Verlag New York/Berlin [26] de Weger, B. M.M, Algorithms for diophantine equations, CWI Tract (1989), Stichting Mathematisch Centrum: Stichting Mathematisch Centrum Amsterdam · Zbl 0687.10013 [27] Wildanger, K., Über das Lösen von Einheiten- und Indexformgleichungen in algebraischen Zahlkörpern mit einer Anwendung auf die Bestimmung aller ganzen Punkte einer Mordellschen Kurve (1997), Technische Universität Berlin: Technische Universität Berlin Berlin · Zbl 0912.11061
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.