Invariants of first order partial differential equations

Abstract

In this paper we introduce the concepts of multiplicity and index of first order partial differential equations. In particular, the concept of multiplicity coincides with the multiplicity of implicit differential equations given by Bruce and Tari in [2]. We also show that these concepts are invariants by smooth equivalences. Following the works Hayakawa, Ishikawa, Izumiya and Yamaguchi on implicit differential equations with first integrals, we introduce a definition of multiplicity for this class of equations