Effective results for unit equations over finitely generated domains

Mathematics – Number Theory

Scientific paper

Rate now

  [ 0.00 ] – not rated yet Voters 0   Comments 0

Details

41 pages

Scientific paper

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.

No associations

LandOfFree

Say what you really think

Search LandOfFree.com for scientists and scientific papers. Rate them and share your experience with other people.

Rating

Effective results for unit equations over finitely generated domains does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Effective results for unit equations over finitely generated domains, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Effective results for unit equations over finitely generated domains will most certainly appreciate the feedback.

Rate now

     

Profile ID: LFWR-SCP-O-414998

  Search
All data on this website is collected from public sources. Our data reflects the most accurate information available at the time of publication.