Linearizability of d-webs, d \geq 4, on two-dimensional manifolds

Abstract

We find d - 2 relative differential invariants for a d-web, d \geq 4, on a two-dimensional manifold and prove that their vanishing is necessary and sufficient for a d-web to be linearizable. If one writes the above invariants in terms of web functions f (x,y) and g_4 (x,y),...,g_d (x,y), then necessary and sufficient conditions for the linearizabilty of a d-web are two PDEs of the fourth order with respect to f and g_4, and d - 4 PDEs of the second order with respect to f and g_4,...,g_d. For d = 4, this result confirms Blaschke's conjecture on the nature of conditions for the linearizabilty of a 4-web. We also give Mathematica codes for testing 4- and d-webs (d > 4) for linearizability and examples of their usage.