Estimate of the number of zeros of an abelian integral depending on a parameter and limit cycles

A. N. Varchenko

Funkcionalʹnyj analiz i ego priloženiâ, Tome 18 (1984) no. 2, pp. 14-25.

Bibliographie

[1] Arnold V.I., “Poterya ustoichivosti avtokolebanii vblizi rezonansa i versalnye deformatsii ekvivariantnykh vektornykh polei”, Funkts. analiz, 11:2 (1977), 1–10 | MR

[2] Arnold V.I., Oleinik O.A., “Topologiya deistvitelnykh algebraicheskikh mnogoobrazii”, Vestn. MGU, seriya matem., 1979, no. 6, 7–17 | MR

[3] Deligne P., Equations differentielles a points singuliers reguliers, Lect. Notes Math., 163, 1970 | DOI | MR | Zbl

[4] Katz N., “Nilpotent connections and the monodromy theorem: applications of a result of Turritin”, Publ. Math. I.H.E.S., 39 (1972), 355–412 | MR

[5] Malgranzh B., “O polinomakh I.N. Bernshteina”, UMN, 29:4 (1974), 81–88 | MR

[6] Khovanskii A.G., “Ob odnom klasse sistem transtsendentnykh uravnenii”, DAN SSSR, 255:4 (1980), 804–807 | MR

[7] Hovansky A., Theorema de Bezout pur les functions de Liouville, Preprint M/81/45 - IHES, IHES, Bures-sur-Ivette, 1981

[8] Gabrielov A.M., “O proektsiyakh poluanaliticheskikh mnozhestv”, Funkts. analiz, 2:4 (1968), 18–30 | MR | Zbl

[9] Varchenko A.N., “Teoremy topologicheskoi ekvisingulyarnosti semeistv algebraicheskikh mnogoobrazii i semeistv polinomialnykh otobrazhenii”, Izv. AN SSSR, seriya matem., 36:5 (1972), 957–1019

[10] Andronov A.A., Leontovich E.A., Gordon I.I., Mauer A.G., Teoriya bifurkatsii dinamicheskikh sistem na ploskosti, Nauka, M., 1967 | MR

[11] Hilbert D., “Mathematische Probleme”, Arch. Math. Phys., 3:1 (1901), 213–237 | MR

[12] Bogdanov P.I., “Bifurkatsiya predelnogo tsikla odnogo semeistva vektornykh polei na ploskosti”, Trudy sem. im. I.G. Petrovskogo, no. 2, Izd. MGU, M., 1976, 23–25 | MR

[13] Khorozov E.I., “Versalnye deformatsii ekvivariantnykh vektornykh polei dlya sluchaev simmetrii poryadka 2 i 3”, Trudy sem. im. I.G. Petrovskogo, no. 5, Izd. MGU, M., 1979, 163–192 | MR

[14] Neishtadt A.I., “Bifurkatsii fazovogo portreta odnoi sistemy uravnenii, voznikayuschei v zadache o potere ustoichivosti avtokolebanii vblizi rezonansa 1:4”, PMM, 42, no. 5, 1978, 830–840 | MR

[15] Ilyashenko Yu.S., “O chisle nulei nekotorykh abelevykh integralov v veschestvennoi oblasti”, Funkts. analiz, 11:4 (1977), 78–79 | MR

[16] Petrov G.S., “O chisle nulei polnykh ellipticheskikh integralov”, Funkts. analiz, 18:2 (1984), 73–74 | MR | Zbl

[17] Malgrange B., “Integrales asymptotiques et monodromie”, Ann. Sclent. Ecole Norm. Super, 7 (1974), 405–430 | DOI | MR | Zbl

[18] Brieskorn E., “Die Monodromie der isolierten Singularitaten von Hyperflächen”, Manuskripta Math., 1970, no. 2, 103–161 | DOI | MR | Zbl

[19] Schmid W., “Variation of Hodge structures: the singularities of period mappings”, Invent. Math., 22 (1973), 211–319 | DOI | MR | Zbl

[20] Landman A., “On the Picard–Lefschetz transformation for algebraic manifolds aquiring general singularities”, Trans. Amer. Math. Soc., 181, July, 1973, 89- 126 | DOI | MR | Zbl

[21] Khironaka X., “Razreshenie osobennostei algebraicheskikh mnogoobrazii nad polyami kharakteristiki nul”, Matematika, 9:6 (1965), 2–70; 10:1 (1966), 3–89 ; 10:2 (1966), 3–58 | MR | MR

[22] Borel A., Lineinye algebraicheskie gruppy, Mir, M., 1972 | MR | Zbl

[23] Grothendieck A., “On the de Rham cohomology of algebraic varieties”, Publ. Math. I.H.E.S., 29 (1966), 95–103 | DOI | MR

[24] Pontryagin L.S., “O dinamicheskikh sistemakh, blizkikh k gamiltonovym”, ZhETF, 4:8 (1934), 57–68 | MR

[25] Khovanskii A.G., “Veschestvennye analiticheskie mnogoobraziya so svoistvom konechnosti i kompleksnye abelevy integraly”, Funkts. analiz, 18:2 (1984), 40–50 | MR

[26] Arnold V.I., Varchenko A.N., Gusein-zade S.M., Osobennosti differentsiruemykh otobrazhenii, t. II, Nauka, M., 1984 | MR