Notes on trimodal singularities

Abstract:

Singularities of functions of modality 1 or 2 were classified by Arnold [3]. Types of those of modality 3 were listed in [14]. Here we show that these are all Newton non-degenerate in the extended sense introduced in [17], and use this fact to obtain explicit regular bases and precise normal forms for right and for contact equivalence. This requires care since the only existing method for non-semiquasihomogeneous germs (using Arnold's “Condition A”) does not apply to all our cases.