Some Phases of Oscillatory Integrals

The first example of a phase is presented for which Arhold's conjecture on the validity of uniform estimates for oscillatory integrals with maximal singularity index is true, while his conjecture on the semicontinuity of the singularity index is false. A rough upper bound for the Milnor number such that the latter conjecture fails is obtained. The corresponding counterexample is simpler than Varchenko's well-known counterexample to Arnold's conjecture on the semicontinuity of the singularity index. This gives hope to decrease codimension and the Milnor number for which the conjecture on the semicontinuity of the singularity index fails.