On the multiplicity of implicit differential equations
In this paper the study of the implicit differential equation (IDE) \[ F(x,y,dy/dx)=0 \] (where \(F\) is an analytic function in \((x,y))\) is motivated by some concrete problems concerning binary differential equations (BDE) of the form: \[ a(x,y)dy^2+2b(x,y)dxdy+c(x,y)dx^2=0. \] The computation of the local multiplicity of a germ of a singularity of the IDE is considered, using some results from a paper of Montaldi and van Straten. In particular, the singularities of the canonical 1-form \(dy-pdx\) are characterized to have multiplicity 1 on the criminant of \(F\), and the set of equations is established for which there is finite multiplicity. For the binary differential equations the authors give an expression for the multiplicity in terms of the discriminant and on the other hand they study BDEs all of whose coefficients vanish at the origin. The multiplicity of many germs of BDEs is computed whose discriminants are simple plane curve singularities.
scientific article; zbMATH DE number 1212358
scientific article