Anomaly of Differential Dimension of Intersections

In [2, Chapter VII (p. 133)], Ritt shows an example of two irreducible differential variety of differential dimension 2 in U whose intersection consists of a single point: the origin. The details below follow Ritt’s exposition using more modern language and notation. The first part is taken from my tutorial paper [3], with some change in notation. That paper provides the basic background needed. A generalization of Ritt’s example to the partial case in U (n > 3), where an irreducible closed set in U of differential dimension n− 1 intersects the hyperplane yn = 0 in the single point (0, . . . , 0), is mentioned in Kolchin [1, p. 194, Ex. 3.]. In general, there is no easy way to show that (0, 0, 0) is a solution of the general component of an irreducible differential polynomial in three variables, much less it is the only one. The first question is the (still unsolved) Ritt Problem and partial solutions involve deep results in differential algebra (for example, the Leading Coefficient Theorem, Prop. 6 on p. 191 and Theorem 4 on p. 172 of Kolchin [1], where the problem is treated with more generality). The second question, that (α, β, γ) is not a solution of the general component, is equally difficult and involves extending differentially the base field to include α, β, γ, translating the variables to x − α, y − β, z − γ, and reducing it to the problem of showing that (0, 0, 0) is not a solution. The only known results are Prop. 7 of p. 192 and Prop. 8 and some special cases, which may be traced to Levi’s Lemma (p. 177) and the Domination Lemma (p. 181) in Kolchin’s text. See also Levi’s Lemma (p. 66) in Ritt [2].