Ramified Integrals, Singularities and Lacunas

I. Picard—Lefschetz—Pham theory and singularity theory.- § 1. Gauss-Manin connection in the homological bundles. Monodromy and variation operators.- § 2. The Picard-Lefschetz formula. The Leray tube operator.- § 3. Local monodromy of isolated singularities of holomorphic functions.- § 4. Intersection form and complex conjugation in the vanishing homology of real singularities in two variables.- § 5. Classification of real and complex singularities of functions.- § 6. Lyashko-Looijenga covering and its generalizations.- § 7. Complements of discriminants of real simple singularities (after E. Looijenga).- § 8. Stratifications. Semialgebraic, semianalytic and subanalytic sets.- § 9. Pham’s formulae.- § 10. Monodromy of hyperplane sections.- § 11. Stabilization of local monodromy and variation of hyperplane sections close to strata of positive dimension (stratified Picard-Lefschetz theory).- § 12. Homology of local systems. Twisted Picard-Lefschetz formulae.- § 13. Singularities of complete intersections and their local monodromy groups.- II. Newton’s theorem on the nonintegrability of ovals.- § 1. Stating the problems and the main results.- § 2. Reduction of the integrability problem to the (generalized) PicardLefschetz theory.- § 3. The element “cap”.- § 4. Ramification of integration cycles close to nonsingular points. Generating functions and generating families of smooth hypersurfaces.- § 5. Obstructions to integrability arising from the cuspidal edges. Proof of Theorem 1.8.- § 6. Obstructions to integrability arising from the asymptotic hyperplanes. Proof of Theorem 1.9.- § 7. Several open problems.- III. Newton’s potential of algebraic layers.- § 1. Theorems of Newton and Ivory.- § 2. Potentials of hyperbolic layers are polynomialin the hyperbolicity domains (after Arnold and Givental).- § 3. Proofs of Main Theorems 1 and 2.- § 4. Description of the small monodromy group.- § 5. Proof of Main Theorem 3.- IV. Lacunas and the local Petrovski$$
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$$ condition for hyperbolic differential operators with constant coefficients.- § 0. Hyperbolic polynomials.- § 1. Hyperbolic operators and hyperbolic polynomials. Sharpness, diffusion and lacunas.- § 2. Generating functions and generating families of wave fronts for hyperbolic operators with constant coefficients. Classification of the singular points of wave fronts.- § 3. Local lacunas close to nonsingular points of fronts and to singularities A2, A3 (after Davydova, Borovikov and Gárding).- § 4. Petrovskii and Leray cycles. The Herglotz-Petrovskii—Leray formula and the Petrovskii condition for global lacunas.- § 5. Local Petrovskii condition and local Petrovskii cycle. The local Petrovskii condition implies sharpness (after Atiyah, Bott and Gárding).- § 6. Sharpness implies the local Petrovskii condition close to discrete-type points of wave fronts of strictly hyperbolic operators.- § 7. The local Petrovskii condition may be stronger than the sharpness close to singular points not of discrete type.- § 8. Normal forms of nonsharpness close to singularities of wave fronts (after A.N. Varchenko).- § 9. Several problems.- V. Calculation of local Petrovski$$
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$$ cycles and enumeration of local lacunas close to real function singularities.- § 1. Main theorems.- § 2. Local lacunas close to singularities from the classification tables.- § 3. Calculation of the even local Petrovskii class.- § 4. Calculation of theodd local Petrovskii class.- § 5. Stabilization of the local Petrovskii classes. Proof of Theorem 1.5.- § 6. Local lacunas close to simple singularities.- § 7. Geometrical criterion for sharpness close to simple singularities.- § 8. A program for counting topologically different morsifications of a real singularity.- § 9. More detailed description of the algorithm.- Appendix: a FORTRAN program searching for the lacunas and enumerating the morsifications of real function singularities.