Effective results for Diophantine equations over finitely generated domains

TL;DR

This paper establishes explicit bounds and effective methods for solving Diophantine equations over finitely generated domains, extending classical results to more general algebraic settings and providing tools for their explicit resolution.

Contribution

It generalizes effective bounds and solution methods for Thue, hyper-, and superelliptic equations to finitely generated domains over Z, including a generalization of Schinzel and Tijdeman's theorem.

Findings

Explicit upper bounds for solutions in finitely generated domains.

Solutions can be determined effectively in principle.

A computable constant C bounds the exponents for which solutions exist.

Abstract

Let A be an arbitrary integral domain of characteristic 0 which is finitely generated over Z. We consider Thue equations $F(x,y)=b$ with unknowns x,y from A and hyper- and superelliptic equations $f(x)=by^m$ with unknowns from A, where the binary form F and the polynomial f have their coefficients in A, where b is a non-zero element from A, and where m is an integer at least 2. Under the necessary finiteness conditions imposed on F,f,m, we give explicit upper bounds for the sizes of x,y in terms of suitable representations for A,F,f,b Our results imply that the solutions of Thue equations and hyper- and superelliptic equations over arbitrary finitely generated domains can be determined effectively in principle. Further, we generalize a theorem of Schinzel and Tijdeman to the effect, that there is an effectively computable constant C such that $f(x)=by^m$ has no solutions in x,y from A with y not 0 or a root of unity if m>C. In our proofs, we use effective results for Thue equations and hyper- and superelliptic equations over number fields and function fields, some effective commutative algebra, and a specialization argument.