Abstract
All components of complements of discriminant varieties of simple real function singularities are explicitly listed. A combinatorial algorithm enumerating the topological types of morsifications of real function singularities is promoted.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
N. A’Campo, Le groupe de monodromie du déploiement des singularités isolées de courbes planes. I, Mathematische Annalen 213 (1975), 1–32.
M. Atiyah, R. Bott and L. Gårding, Lacunas for hyperbolic differential operators with constant coefficients. I, II, Acta Mathematica 124 (1970), 109–189, 131 (1973), 145–206.
V. I. Arnold, On some topological invariants of algebraic functions, Transactions of the Moscow Mathematical Society 21 (1970), 30–52.
V. I. Arnold, A. N. Varchenko and S. M. Gusein-Zade, Singularities of Differentiable Maps. Vol. 1, Nauka, Moscow, 1982; English translation: Modern Birkhäuser Classics, Birkhäuser/Springer, New York, 2012.
V. I. Arnold, A. N. Varchenko, and S. M. Gusein-Zade, Singularities of Differentiable Maps. Vol. 2, Nauka, Moscow, 1984; English translation: Monographs in Mathematics, Vol. 83, Birkhäuser, Boston, MA, 1988.
L. Gårding, Sharp fronts of paired oscillatory integrals, Kyoto University. Research Institute for Mathematical Sciences. Publications 12 (1977), 53–68.
S. M. Gusein-Zade, Intersection matrices for certain singularities of functions of two variables, Functional Analysis and its Applications 8 (1974), 10–13.
L. Hörmander, Fourier integral operators. I, Acta Mathematica 127 (1971), 79–183.
V. M. Kharlamov, Rigid isotopy classification of real plane curves of degree 5, Functional Analysis and its Applications 15 (1981), 73–74.
J. Leray, Un prolongement de la transformation de Laplace qui transforme la solution unitaire d’un opérateur hyperbolique en sa solution élémentaire (Probleme de Cauchy, IV), Bulletin de la Société Mathématique de France 90 (1962), 39–156.
E. Looijenga, The complement of the bifurcation variety of a simple singularity, Inventiones Mathematicae 23 (1974), 105–116.
E. Looijenga, The discriminant of a real simple singularity, Compositio Mathematica 37 (1978), 51–62.
J. Milnor, Singular Points of Complex Hypersurfaces, Annals of Mathematics Studies, Vol. 61, Princeton University Press; Tokyo University Press, 1968.
I. G. Petrovsky, On the diffusion of waves and the lacunas for hyperbolic equations, Matematicheskii Sbornik, 17 (1945), 289–370.
G. M. Polotovskii, Connection between the rigid isotopy class of a fifth-order nonsingular curve in ℝP2and its disposition with respect to a line, Functional Analysis and its Applications 20 (1986), 330–332.
V. A. Rokhlin, Complex topological characteristics of real algebraic curves, Russian Mathematical Surveys 33 (1978), 85–98.
V. D. Sedykh, On the topology of wave fronts in spaces of low dimension, Izvestiya, Mathematics 76 (2012), 375–418.
V. D. Sedykh, Mathematical Methods of Catastrophe Theory, MCCME, Moscow, 2021.
A. N. Varchenko, On normal forms of nonsmoothness of solutions of hyperbolic equations, Mathematics of the USSR-Izvestiya 30 (1988), 615–628.
V. A. Vassiliev, Stable cohomology of complements to the discriminants of deformations of singularities of smooth functions, Journal of Soviet Mathematics 52 (1990), 3217–3230.
V. A. Vassiliev, Complements of Discriminants of Smooth Maps: Topology and Applications, Translations of Mathematical Monographs, Vol. 98, American Mathematical Society, Providence, RI, 1992.
V. A. Vassiliev, Applied Picard–Lefschetz Theory, Mathematical Surveys and Monographs, Vol. 97, American Mathematical Society, Providence, RI, 2002.
V. A. Vassiliev, New examples of irreducible local diffusion of hyperbolic PDE’s, SIGMA 16 (2020), Article no. 9.
V. A. Vassiliev, morscounX90, A program counting real morsifications of isolated real function singularities of corank ≤ 2, https://drive.google.com/file/d/1RtiqiuBBNYtf218zNRK8SIoFhA6GRb38/view?usp=sharing
O. Ya. Viro, Progress in the topology of real algebraic varieties over the last six years, Russian Mathematical Surveys 41 (1986), 55–82.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Vassiliev, V.A. Complements of discriminants of simple real function singularities. Isr. J. Math. (2024). https://doi.org/10.1007/s11856-024-2627-8
Received:
Revised:
Published:
DOI: https://doi.org/10.1007/s11856-024-2627-8