Auxiliary Results and Tools

István Gaál
Additional contact information
István Gaál: University of Debrecen, Institute of Mathematics

Chapter Chapter 2 in Diophantine Equations and Power Integral Bases, 2019, pp 13-24 from Springer

Abstract: Abstract In this chapter we summarize the basic tools we use all over the book to solve our Diophantine equations. The equations are usually reduced to the so-called unit equations Unit equation of type α u + β v = 1 $$\displaystyle \alpha u+\beta v =1 $$ (cf. Eq. (2.5)) with given algebraic α, β, where u, v are unknown units in a number field. These units are written as a power product of the generators of the unit group and the unknown exponents are to be determined. Baker’s method (Sect. 2.1) is used to give an initial upper bound for the unknowns, which is of magnitude 1018 for the simplest Thue equations but 10100 is also not unusual for more complicated equations. We apply numerical Diophantine approximation techniques based on the LLL basis reduction algorithm (Sect. 2.2) to reduce these bounds. The reduced bounds are usually between 100 and 1000. These reduced bounds are quite modest; however, if there are more than four to five unknown exponents, it is already impossible to test directly all possible exponents with absolute values under the reduced bound. Hence it is crucial to apply some efficient enumeration methods (Sect. 2.3) to overcome this difficulty. In fact we observe that the majority of computations are spent by proving that there are no “large” solutions. According to our experience the solutions of the equations we consider are usually “small” and can be found sometimes quite easily and fast.

Date: 2019
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-030-23865-0_2

Ordering information: This item can be ordered from
http://www.springer.com/9783030238650

DOI: 10.1007/978-3-030-23865-0_2

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().