New examples (and counterexamples) of complete finite-rank differential varieties

Differential algebraic geometry seeks to extend the results of its algebraic counterpart to objects defined by differential equations. Many notions, such as that of a projective algebraic variety, have close differential analogues but their behavior can vary in interesting ways. Workers in both diff...

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Published in:Communications in algebra 2017-07, Vol.45 (7), p.3137-3149
Main Author: Simmons, William D.
Format: Article
Language:English
Subjects:
Algebra
Complete variety
Completeness
Criteria
differential algebra
differential algebraic geometry
Differential equations
Differential geometry
model theory
Reviewing
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description Differential algebraic geometry seeks to extend the results of its algebraic counterpart to objects defined by differential equations. Many notions, such as that of a projective algebraic variety, have close differential analogues but their behavior can vary in interesting ways. Workers in both differential algebra and model theory have investigated the property of completeness of differential varieties. After reviewing their results, we extend that work by proving several versions of a "differential valuative criterion" and using them to give new examples of complete differential varieties. We conclude by analyzing the first examples of incomplete, finite-rank projective differential varieties, demonstrating a clear difference from projective algebraic varieties.
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subjects Algebra
Complete variety
Completeness
Criteria
differential algebra
differential algebraic geometry
Differential equations
Differential geometry
model theory
Reviewing
title New examples (and counterexamples) of complete finite-rank differential varieties
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