The problem of defining the singular points of quasi-linear differential-algebraic systems

Abstract

We introduce here an algebraic characterization of fixed singularities of a complex polynomial quasi-linear differential-algebraic system A( t, y) y′ − b( t, y) = 0. We begin with explaining how to represent it by using a finite number of saturated modules, defined over irreducible varieties, named their constraint varieties. A projection-elimination process, inspired from Reich (1991), yields those modules, whose main property is to be invariant under the adding of the derivatives of the equations of their constraint variety. We call them differentially stable systems. Each has a proper index, the number of derivations needed to get it. For one of those modules, we define the fixed singular points as those points of the constraint variety which reduce the rank of the matrix of the differential part. Our definition is shown to be more intrinsic than the one based on differential algebra, based on characteristic sets to represent the solutions: the singular sets we obtain are included in these provided by differential algebra methods. The second part deals with the geometry of singular points, which are divided into three subsets. We show in particular that the points where the constraint variety is not a manifold can be investigated by the introduction of their tangent cone, consisting of the vectors that are tangent to the variety. We eventually give some examples proving that the set of singular points is not always the vanishing set of a single polynomial, unlike in the case without constraint, where the determinant plays this role.