Invariants of forth order linear differential operators (Q2225833)

This paper continues the previous papers by the authors, where they analyzed the equivalence of \(k\)-th order linear differential operators acting on a line bundle over smooth \(n\)-dimensional manifold. There are two opposite cases: when the symbol has fixed type, and when type of the symbol contains maximum functional parameters modulo the equivalence group. In the first case they solved the equivalence problem via the Wagner connection. In the second case, provided \(n\ge2\) and \(k\ge3\) (the case \(k=2\) was successfully treated earlier) the invariants of the symbol provide \(n\) independent order zero invariants for all cases except \((k,n)=(2,3)\), \((2,4)\) or \((3,3)\). The first of the exceptional cases was considered by the authors earlier, and this paper is dedicated to the second exceptional case \(n=2\), \(k=4\). There is one rational invariant of order zero obtained from the symbol (binary quadric) and they construct one more invariant of order 1, allowing to solve the equivalence problem of general \(4\)-th order differential operators on surfaces.