Let A be a commutative domain containing Z which is finitely generated as a
Z-algebra, and let a,b,c be non-zero elements of A. It follows from work of
Siegel, Mahler, Parry and Lang that the equation (*) ax+by=c has only finitely
many solutions in elements x,y of the unit group A* of A, but the proof
following from their arguments is ineffective. Using linear forms in logarithms
estimates of Baker and Coates, in 1979 Gy\H{o}ry gave an effective proof of
this finiteness result, in the special case that A is the ring of S-integers of
an algebraic number field. Some years later, Gy\H{o}ry extended this to a
restricted class of finitely generated domains A, containing transcendental
elements. In the present paper, we give an effective finiteness proof for the
number of solutions of (*) for arbitrary domains A finitely generated over Z.
In fact, we give an explicit upper bound for the `sizes' of the solutions x,y,
in terms of defining parameters for A,a,b,c. In our proof, we use already
existing effective finiteness results for two variable S-unit equations over
number fields due to Gy\H{o}ry and Yu and over function fields due to Mason, as
well as an explicit specialization argument.Comment: 41 page