Speaker:
William Simmons, UIC - University of Illinois at Chicago
Identifying complete differential varieties abstract:
In classical algebraic geometry, the geometric role of compactness is played by the property of completeness: if V is an algebraic variety, then V is complete if for every algebraic variety W the projection V ×W → W is a closed map with respect to the Zariski topology. The fundamental theorem of elimination theory asserts that projective varieties are complete. What happens with differential varieties, i.e., solution sets of differential polynomial equations over differential fields? We discuss several approaches to the problem, with our main focus being a positive quantifier elimination test of van den Dries that was adapted to a differential valuative criterion by Pong. We also touch on completeness in the closely related category of algebraic D-varieties, introduced by Buium and studied model theoretically by Pillay and Kowalski.