Realisations of Non v ‐Sufficient Jets with value in R p , p > 1

The purpose of this paper is to show that if a jet cue οεJ r ( n, p ), n ⩾ p, p > 1, is not v ‐sufficient in C r+1 , there exists an infinite sequence ( f i ) i εN * of realisations of ο with mutually non‐homeomorphic germs of varietiesf i − 1 { 0 } . Bochnak and Kuo [ 2, 5 ] showed it when p = 1 and thought that the same argument slightly modified can be used in the case p ⩾ 2 [ 7 , p. 225]. But when n ⩾ p + 2, p > 1, we have to proceed differently. Moreover, it is necessary to prove separately the result when n = p and n = p + 1. About C 0 ‐sufriciency and p > 1, Brodersen [ 3 , p. 168] showed a similar theorem.