Submission date: 31 October 2014.

Abstract:

Motivated by the effective bounds found in [E. Hrushovski and A. Pillay, “Effective bounds for the number of transcendental points on subvarieties of semi-abelian varieties”, AJM 122(3), 2000] for ordinary differential equations, we prove an effective version of uniform bounding for partial differential fields with commuting derivations. More precisely, we provide an upper bound for the size of finite solution sets of partial differential polynomial equations in terms of data explicitly given in the equations and independent of parameters. Our methods also produce an upper bound for the degree of the Zariski closure of solution sets, whether they are finite or not.

Mathematics Subject Classification: 03C60, 12H05, 12H20

Keywords and phrases: differential algebra, prolongation spaces

Full text arXiv 1411.0029: pdf, ps.