Complete sets in differentially closed fields
Using basic model theory and differential algebraic geometry, the author studies complete sets in the frame of differential algebraic geometry. Let \((F, \delta)\) be a fixed differentialy closed field of characteristic \(0\), with the field of constants \(C\). All \(\delta\)-varieties are identified with the sets of their \(F\)-points. A \(\delta\)-closed set \(X\) in the projective space \(\mathbb{P}^n\) over \(F\) is \(\delta\)-complete if the second projection \(p_2 : X \times Y \to Y\) is a \(\delta\)-closed map for every quasi-projective \(\delta\)-variety \(Y\). By contrast with the situation in algebraic geometry, the author shows that every \(\delta\)-complete set is affine and is definably isomorphic to a \(\delta\)-complete subset of the affine line. Next he obtains a valuative criterion for \(\delta\)-completeness, and, using this criterion, he obtains Kolchin's result on \(\delta\)-completeness of \(\mathbb{P}^n(C)\) and finds a new family of \(\delta\)-complete sets.
scientific article; zbMATH DE number 1428250
scientific article