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December 2021 Calculating “small” solutions of inhomogeneous relative Thue inequalities
István Gaál
Funct. Approx. Comment. Math. 65(2): 141-156 (December 2021). DOI: 10.7169/facm/1876


Thue equations and their relative and inhomogeneous extensions are well known in the literature. There exist methods, usually tedious methods, for the complete resolution of these equations. On the other hand our experiences show that such equations usually do not have extremely large solutions. Therefore in several applications it is useful to have a fast algorithm to calculate the “small” solutions of these equations. Under “small” solutions we mean the solutions, say, with absolute values or sizes $\leq 10^{100}$. Such algorithms were formerly constructed for Thue equations, relative Thue equations. The relative and inhomogeneous Thue equations have applications in solving index form equations and certain resultant form equations. It is also known that certain “totally real” relative Thue equations can be reduced to absolute Thue equations (equations over $\mathbb{Z}$).

As a common generalization of the above results, in our paper we develop a fast algorithm for calculating “small” solutions (say with sizes $\leq 10^{100}$) of inhomogeneous relative Thue equations, more exactly of certain inequalities that generalize those equations. We shall show that in the “totally real” case these can similarly be reduced to absolute inhomogeneous Thue inequalities. We also give an application to solving certain resultant equations in the relative case.


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István Gaál. "Calculating “small” solutions of inhomogeneous relative Thue inequalities." Funct. Approx. Comment. Math. 65 (2) 141 - 156, December 2021.


Published: December 2021
First available in Project Euclid: 13 October 2021

Digital Object Identifier: 10.7169/facm/1876

Primary: 11Y50
Secondary: 11D41 , 11D57 , 11D59

Keywords: calculating solutions , Inequalities‎ , inhomogeneous Thue equations , LLL reduction , relative Thue equations , resultant equations , Thue equations

Rights: Copyright © 2021 Adam Mickiewicz University


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Vol.65 • No. 2 • December 2021
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