Typical differential dimension of the intersection of linear differential algebraic groups

Let % be a universal differential field of characteristic zero. Let “y; and “yz be two irreducible closed (in the differential Zariski topology) subsets of %!n of differential dimension rr and rz , respectively. In general, it is not true that every irreducible component of the intersection Y1 n $$, has a differential dimension 3 rr + ~a - n. Indeed Ritt [3, p. 1331 has an example of an irreducible closed subset of &s of differential dimension 2 that intersects the hyperplane defined by the equation y3 = 0 at the single point (0, 0,O). In this paper (Theorem 4.1 and corollaries) we show that such abnormality does not arise in two special cases: namely, when Vr and Vs are differential algebraic subgroups of either G,” or GL(n). Here G,” denotes the differential algebraic group whose underlying set is 9P and GL(n) denotes the general linear group of n