Comments on differential invariants.

*(English)*Zbl 0621.12018
Infinite dimensional groups with applications, Publ., Math. Sci. Res. Inst. 4, 355-370 (1985).

This paper is intended to stimulate research in the undeveloped theory of differential invariants by bringing together a number of the scattered results now available and listing research problems.

One example is the interesting and far from obvious set of relative differential invariants of a linear differential equation \(L(y)=0\) with independent variable \(t\) under the transformation \(y_1=\lambda(t)y\), \(t_1=\mu(t)\). These invariants were found by E. L. Wilczynski and have been studied by H. Morikawa. Following work of G. H. Halphen they are used to study curves in projective space. Let \(P^K\) denote \(K\)-dimensional projective space over the complex numbers \(\mathbb{C}\). The author shows that the linear equivalence classes of curves in \(P^{(n-1)}\) which are not contained in any \(P^{(n-2)}\) correspond bijectively with the equivalence classes under the transformations above of equations \(L(y)=0\) of order \(n\).

In the direction of a more general theory he considers a differential algebra \(A\) over a differential field \(K\) and a reductive algebraic group \(G\) defined over \(\mathbb{C}\). Suppose that \(G(\mathbb{C})\) acts on \(A\) by differential \(K\)-algebra automorphisms, and let \(A^{G(\mathbb{C})}\) be the subset of \(A\) fixed under \(G(\mathbb{C})\). One wishes to know what properties of \(A\) are inherited by \(A^{G(\mathbb{C})}\). Suppose that \(A\) has the property that every differential ideal of \(A\) is differentially finitely generated.

In Example 4, which is a modification of an example due to J. F. Ritt, it is shown that \(A^{G(\mathbb{C})}\) need not inherit this property. Following the work of Ritt it is, however, natural to consider differential rings in which every radical differential ideal is the radical of a finitely generated differential ideal. The author calls such a ring an r-d-N-ring. He proves under quite general conditions that if \(A\) is r-d-N, then so is \(A^{G(\mathbb{C})}\). The case of Example 4 is subsumed under this theorem.

There is much more. The paper closes with the list of research problems mentioned above and a list of references, each of which is briefly described.

Since the tragic disappearance of the author in Chile, this paper remains a program for others to carry out.

For the entire collection see [Zbl 0577.00010].

One example is the interesting and far from obvious set of relative differential invariants of a linear differential equation \(L(y)=0\) with independent variable \(t\) under the transformation \(y_1=\lambda(t)y\), \(t_1=\mu(t)\). These invariants were found by E. L. Wilczynski and have been studied by H. Morikawa. Following work of G. H. Halphen they are used to study curves in projective space. Let \(P^K\) denote \(K\)-dimensional projective space over the complex numbers \(\mathbb{C}\). The author shows that the linear equivalence classes of curves in \(P^{(n-1)}\) which are not contained in any \(P^{(n-2)}\) correspond bijectively with the equivalence classes under the transformations above of equations \(L(y)=0\) of order \(n\).

In the direction of a more general theory he considers a differential algebra \(A\) over a differential field \(K\) and a reductive algebraic group \(G\) defined over \(\mathbb{C}\). Suppose that \(G(\mathbb{C})\) acts on \(A\) by differential \(K\)-algebra automorphisms, and let \(A^{G(\mathbb{C})}\) be the subset of \(A\) fixed under \(G(\mathbb{C})\). One wishes to know what properties of \(A\) are inherited by \(A^{G(\mathbb{C})}\). Suppose that \(A\) has the property that every differential ideal of \(A\) is differentially finitely generated.

In Example 4, which is a modification of an example due to J. F. Ritt, it is shown that \(A^{G(\mathbb{C})}\) need not inherit this property. Following the work of Ritt it is, however, natural to consider differential rings in which every radical differential ideal is the radical of a finitely generated differential ideal. The author calls such a ring an r-d-N-ring. He proves under quite general conditions that if \(A\) is r-d-N, then so is \(A^{G(\mathbb{C})}\). The case of Example 4 is subsumed under this theorem.

There is much more. The paper closes with the list of research problems mentioned above and a list of references, each of which is briefly described.

Since the tragic disappearance of the author in Chile, this paper remains a program for others to carry out.

For the entire collection see [Zbl 0577.00010].

Reviewer: Richard M. Cohn

##### MSC:

12H05 | Differential algebra |

34G10 | Linear differential equations in abstract spaces |

20G20 | Linear algebraic groups over the reals, the complexes, the quaternions |