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Introduction. I. Picard-Lefschetz-Pham theory and singularity theory. II. Newton's theorem on the nonintegrability of ovals. III. Newton's potential of algebraic layers. IV. Lacunas and the local Petrovskii condition for hyperbolic differential operators with constant coefficients. V. Calculation of local Petrovskii cycles and enumeration of local lacunas close to real function singularities. Appendix: a FORTRAN program searching for the lacunas and enumerating the morsifications of real function singularities. Bibliography. Index.