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Ying Chen; Luis Renato G. Dias; Kiyoshi Takeuchi; Mihai Tibăr
Invertible polynomial mappings via Newton non-degeneracy
(Applications polynomiales inversibles et non-dégénérescence des polyèdres de Newton)
Annales de l'institut Fourier, 64 no. 5 (2014), p. 1807-1822, doi: 10.5802/aif.2897
Article PDF | Analyses MR 3330924 | Zbl 06387324
Class. Math.: 14D06, 58K05, 57R45, 14P10, 32S20, 58K15
Mots clés: applications polynomiales réelles ou complexes, ensemble de bifurcation, problème Jacobien, polyèdre de Newton, regularité à l’infini

Résumé - Abstract

On démontre une condition suffisante pour le problème Jacobien dans le contexte des applications polynomiales réelles, complexes ou mixtes. Ceci résulte de l’étude de l’ensemble de bifurcation d’une application soumise à une nouvelle condition de non-dégénérescence par rapport aux polyèdres de Newton à l’infini.

Bibliographie

[1] Andrzej Białynicki-Birula & Maxwell Rosenlicht, “Injective morphisms of real algebraic varieties”, Proc. Amer. Math. Soc. 13 (1962), p. 200-203 Article |  MR 140516 |  Zbl 0107.14602
[2] Carles Bivià-Ausina, “Injectivity of real polynomial maps and Łojasiewicz exponents at infinity”, Math. Z. 257 (2007) no. 4, p. 745-767 Article |  MR 2342551 |  Zbl 1183.14076
[3] S. A. Broughton, On the topology of polynomial hypersurfaces, Singularities, Part 1 (Arcata, Calif., 1981), Proc. Sympos. Pure Math. 40, Amer. Math. Soc., 1983, p. 167–178  MR 713056 |  Zbl 0526.14010
[4] S. A. Broughton, “Milnor numbers and the topology of polynomial hypersurfaces”, Invent. Math. 92 (1988) no. 2, p. 217-241 Article |  MR 936081 |  Zbl 0658.32005
[5] Ying Chen, L. R. G. Dias & M. Tibăr, “On Newton non-degeneracy of polynomial mappings”, arXiv:1207.1612
[6] Ying Chen & Mihai Tibăr, “Bifurcation values and monodromy of mixed polynomials”, Math. Res. Lett. 19 (2012) no. 1, p. 59-79 Article |  MR 2923176 |  Zbl 1274.14006
[7] Sławomir Cynk & Kamil Rusek, “Injective endomorphisms of algebraic and analytic sets”, Ann. Polon. Math. 56 (1991) no. 1, p. 29-35  MR 1145567 |  Zbl 0761.14005
[8] L. R. G. Dias, M. A. S. Ruas & M. Tibăr, “Regularity at infinity of real mappings and a Morse-Sard theorem”, J. Topol. 5 (2012) no. 2, p. 323-340 Article |  MR 2928079 |  Zbl 1248.14014
[9] Alan H. Durfee, Five definitions of critical point at infinity, Singularities (Oberwolfach, 1996), Progr. Math. 162, Birkhäuser, 1998, p. 345–360  MR 1652481 |  Zbl 0919.32021
[10] Arno van den Essen, Polynomial automorphisms and the Jacobian conjecture, Progress in Mathematics 190, Birkhäuser Verlag, Basel, 2000 Article |  Zbl 0962.14037
[11] Alexander Esterov & Kiyoshi Takeuchi, “Motivic Milnor fibers over complete intersection varieties and their virtual Betti numbers”, Int. Math. Res. Not. IMRN (2012) no. 15, p. 3567-3613 Article |  MR 2959042 |  Zbl 1250.32025
[12] Terence Gaffney, “Fibers of polynomial mappings at infinity and a generalized Malgrange condition”, Compositio Math. 119 (1999) no. 2, p. 157-167 Article |  MR 1723126 |  Zbl 0945.32013
[13] A. Grothendieck, “Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. I”, Inst. Hautes Études Sci. Publ. Math. (1964) no. 20 Numdam |  MR 219538 |  Zbl 0135.39701
[14] H. V. Hà & D. T. Lê, “Sur la topologie des polynômes complexes”, Acta Math. Vietnam 9 (1984) no. 1, p. 21-32  Zbl 0597.32005
[15] Zbigniew Jelonek, “Testing sets for properness of polynomial mappings”, Math. Ann. 315 (1999) no. 1, p. 1-35 Article |  MR 1717542 |  Zbl 0946.14039
[16] Zbigniew Jelonek, On asymptotic critical values and the Rabier theorem, Geometric singularity theory, Banach Center Publ. 65, Polish Acad. Sci., 2004, p. 125–133 Article |  MR 2104342 |  Zbl 1160.58311
[17] K. Kurdyka, P. Orro & S. Simon, “Semialgebraic Sard theorem for generalized critical values”, J. Differential Geom. 56 (2000) no. 1, p. 67-92  MR 1863021 |  Zbl 1067.58031
[18] A. Kushnirenko, “Polyèdres de Newton et nombres de Milnor”, Invent. Math. 32 (1976), p. 1-31 Article |  Zbl 0328.32007
[19] Yutaka Matsui & Kiyoshi Takeuchi, “Monodromy zeta functions at infinity, Newton polyhedra and constructible sheaves”, Math. Z. 268 (2011) no. 1-2, p. 409-439 Article |  MR 2805442 |  Zbl 1264.14005
[20] András Némethi & Alexandru Zaharia, “On the bifurcation set of a polynomial function and Newton boundary”, Publ. Res. Inst. Math. Sci. 26 (1990) no. 4, p. 681-689 Article |  MR 1081511 |  Zbl 0736.32024
[21] András Némethi & Alexandru Zaharia, “Milnor fibration at infinity”, Indag. Math. (N.S.) 3 (1992) no. 3, p. 323-335 Article |  MR 1186741 |  Zbl 0806.57021
[22] T. T. Nguyen, “Bifurcation set, $M$-tameness, asymptotic critical values and Newton polyhedrons”, Kodai Math. J. 36 (2013) no. 1, p. 77-90 Article |  MR 3043400 |  Zbl 1266.32036
[23] Mutsuo Oka, Non-degenerate complete intersection singularity, Actualités Mathématiques. [Current Mathematical Topics], Hermann, Paris, 1997  MR 1483897 |  Zbl 0930.14034
[24] Mutsuo Oka, “Topology of polar weighted homogeneous hypersurfaces”, Kodai Math. J. 31 (2008) no. 2, p. 163-182 Article |  MR 2435890 |  Zbl 1149.14031
[25] Mutsuo Oka, “Non-degenerate mixed functions”, Kodai Math. J. 33 (2010) no. 1, p. 1-62 Article |  MR 2732230 |  Zbl 1195.14061
[26] Adam Parusiński, “On the bifurcation set of complex polynomial with isolated singularities at infinity”, Compositio Math. 97 (1995) no. 3, p. 369-384 Numdam |  MR 1353280 |  Zbl 0840.32007
[27] Sergey Pinchuk, “A counterexample to the strong real Jacobian conjecture”, Math. Z. 217 (1994) no. 1, p. 1-4 Article |  MR 1292168 |  Zbl 0874.26008
[28] Patrick J. Rabier, “Ehresmann fibrations and Palais-Smale conditions for morphisms of Finsler manifolds”, Ann. of Math. (2) 146 (1997) no. 3, p. 647-691 Article |  MR 1491449 |  Zbl 0919.58003
[29] Dirk Siersma & Mihai Tibăr, “Singularities at infinity and their vanishing cycles”, Duke Math. J. 80 (1995) no. 3, p. 771-783 Article |  MR 1370115 |  Zbl 0871.32024
[30] Masakazu Suzuki, “Propriétés topologiques des polynômes de deux variables complexes, et automorphismes algébriques de l’espace ${\bf C}^{2}$”, J. Math. Soc. Japan 26 (1974), p. 241-257 Article |  MR 338423 |  Zbl 0276.14001
[31] Mihai Tibăr, Regularity at infinity of real and complex polynomial functions, Singularity theory (Liverpool, 1996), London Math. Soc. Lecture Note Ser. 263, Cambridge Univ. Press, 1999, p. xx, 249–264  MR 1709356 |  Zbl 0930.58005
[32] Mihai Tibăr, Polynomials and vanishing cycles, Cambridge Tracts in Mathematics 170, Cambridge University Press, Cambridge, 2007 Article |  MR 2360234 |  Zbl 1126.32026
[33] Mihai Tibăr & Alexandru Zaharia, “Asymptotic behaviour of families of real curves”, Manuscripta Math. 99 (1999) no. 3, p. 383-393 Article |  MR 1702581 |  Zbl 0965.14012
[34] Jean-Louis Verdier, “Stratifications de Whitney et théorème de Bertini-Sard”, Invent. Math. 36 (1976), p. 295-312 Article |  MR 481096 |  Zbl 0333.32010
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