Invertible polynomial mappings via Newton non-degeneracy
[Applications polynomiales inversibles et non-dégénérescence des polyèdres de Newton]
Chen, Ying ; Dias, Luis Renato G. ; Takeuchi, Kiyoshi ; Tibăr, Mihai
Annales de l'Institut Fourier, Tome 64 (2014), p. 1807-1822 / Harvested from Numdam

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.

We prove a sufficient condition for the Jacobian problem in the setting of real, complex and mixed polynomial mappings. This follows from the study of the bifurcation locus of a mapping subject to a new Newton non-degeneracy condition.

Publié le : 2014-01-01
DOI : https://doi.org/10.5802/aif.2897
Classification:  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
@article{AIF_2014__64_5_1807_0,
     author = {Chen, Ying and Dias, Luis Renato G. and Takeuchi, Kiyoshi and Tib\u ar, Mihai},
     title = {Invertible polynomial mappings via Newton non-degeneracy},
     journal = {Annales de l'Institut Fourier},
     volume = {64},
     year = {2014},
     pages = {1807-1822},
     doi = {10.5802/aif.2897},
     zbl = {06387324},
     mrnumber = {3330924},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_2014__64_5_1807_0}
}
Chen, Ying; Dias, Luis Renato G.; Takeuchi, Kiyoshi; Tibăr, Mihai. Invertible polynomial mappings via Newton non-degeneracy. Annales de l'Institut Fourier, Tome 64 (2014) pp. 1807-1822. doi : 10.5802/aif.2897. http://gdmltest.u-ga.fr/item/AIF_2014__64_5_1807_0/

[1] Białynicki-Birula, Andrzej; Rosenlicht, Maxwell Injective morphisms of real algebraic varieties, Proc. Amer. Math. Soc., Tome 13 (1962), pp. 200-203 | Article | MR 140516 | Zbl 0107.14602

[2] Bivià-Ausina, Carles Injectivity of real polynomial maps and Łojasiewicz exponents at infinity, Math. Z., Tome 257 (2007) no. 4, pp. 745-767 | Article | MR 2342551 | Zbl 1183.14076

[3] Broughton, S. A. On the topology of polynomial hypersurfaces, Singularities, Part 1 (Arcata, Calif., 1981), Amer. Math. Soc., Providence, RI (Proc. Sympos. Pure Math.) Tome 40 (1983), pp. 167-178 | MR 713056 | Zbl 0526.14010

[4] Broughton, S. A. Milnor numbers and the topology of polynomial hypersurfaces, Invent. Math., Tome 92 (1988) no. 2, pp. 217-241 | Article | MR 936081 | Zbl 0658.32005

[5] Chen, Ying; Dias, L. R. G.; Tibăr, M. On Newton non-degeneracy of polynomial mappings (arXiv:1207.1612)

[6] Chen, Ying; Tibăr, Mihai Bifurcation values and monodromy of mixed polynomials, Math. Res. Lett., Tome 19 (2012) no. 1, pp. 59-79 | Article | MR 2923176 | Zbl 1274.14006

[7] Cynk, Sławomir; Rusek, Kamil Injective endomorphisms of algebraic and analytic sets, Ann. Polon. Math., Tome 56 (1991) no. 1, pp. 29-35 | MR 1145567 | Zbl 0761.14005

[8] Dias, L. R. G.; Ruas, M. A. S.; Tibăr, M. Regularity at infinity of real mappings and a Morse-Sard theorem, J. Topol., Tome 5 (2012) no. 2, pp. 323-340 | Article | MR 2928079 | Zbl 1248.14014

[9] Durfee, Alan H. Five definitions of critical point at infinity, Singularities (Oberwolfach, 1996), Birkhäuser, Basel (Progr. Math.) Tome 162 (1998), pp. 345-360 | MR 1652481 | Zbl 0919.32021

[10] Van Den Essen, Arno Polynomial automorphisms and the Jacobian conjecture, Birkhäuser Verlag, Basel, Progress in Mathematics, Tome 190 (2000), pp. xviii+329 | Article | Zbl 0962.14037

[11] Esterov, Alexander; Takeuchi, Kiyoshi Motivic Milnor fibers over complete intersection varieties and their virtual Betti numbers, Int. Math. Res. Not. IMRN (2012) no. 15, pp. 3567-3613 | Article | MR 2959042 | Zbl 1250.32025

[12] Gaffney, Terence Fibers of polynomial mappings at infinity and a generalized Malgrange condition, Compositio Math., Tome 119 (1999) no. 2, pp. 157-167 | Article | MR 1723126 | Zbl 0945.32013

[13] Grothendieck, A. Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. I, Inst. Hautes Études Sci. Publ. Math. (1964) no. 20, pp. 259 | Numdam | MR 219538 | Zbl 0136.15901 | Zbl 0135.39701

[14] Hà, H. V.; Lê, D. T. Sur la topologie des polynômes complexes, Acta Math. Vietnam, Tome 9 (1984) no. 1, pp. 21-32 | Zbl 0597.32005

[15] Jelonek, Zbigniew Testing sets for properness of polynomial mappings, Math. Ann., Tome 315 (1999) no. 1, pp. 1-35 | Article | MR 1717542 | Zbl 0946.14039

[16] Jelonek, Zbigniew On asymptotic critical values and the Rabier theorem, Geometric singularity theory, Polish Acad. Sci., Warsaw (Banach Center Publ.) Tome 65 (2004), pp. 125-133 | Article | MR 2104342 | Zbl 1160.58311

[17] Kurdyka, K.; Orro, P.; Simon, S. Semialgebraic Sard theorem for generalized critical values, J. Differential Geom., Tome 56 (2000) no. 1, pp. 67-92 http://projecteuclid.org/getRecord?id=euclid.jdg/1090347525 | MR 1863021 | Zbl 1067.58031

[18] Kushnirenko, A. Polyèdres de Newton et nombres de Milnor, Invent. Math., Tome 32 (1976), pp. 1-31 | Article | Zbl 0328.32007

[19] Matsui, Yutaka; Takeuchi, Kiyoshi Monodromy zeta functions at infinity, Newton polyhedra and constructible sheaves, Math. Z., Tome 268 (2011) no. 1-2, pp. 409-439 | Article | MR 2805442 | Zbl 1264.14005

[20] Némethi, András; Zaharia, Alexandru On the bifurcation set of a polynomial function and Newton boundary, Publ. Res. Inst. Math. Sci., Tome 26 (1990) no. 4, pp. 681-689 | Article | MR 1081511 | Zbl 0736.32024

[21] Némethi, András; Zaharia, Alexandru Milnor fibration at infinity, Indag. Math. (N.S.), Tome 3 (1992) no. 3, pp. 323-335 | Article | MR 1186741 | Zbl 0806.57021

[22] Nguyen, T. T. Bifurcation set, M-tameness, asymptotic critical values and Newton polyhedrons, Kodai Math. J., Tome 36 (2013) no. 1, pp. 77-90 | Article | MR 3043400 | Zbl 1266.32036

[23] Oka, Mutsuo Non-degenerate complete intersection singularity, Hermann, Paris, Actualités Mathématiques. [Current Mathematical Topics] (1997), pp. viii+309 | MR 1483897 | Zbl 0930.14034

[24] Oka, Mutsuo Topology of polar weighted homogeneous hypersurfaces, Kodai Math. J., Tome 31 (2008) no. 2, pp. 163-182 | Article | MR 2435890 | Zbl 1149.14031

[25] Oka, Mutsuo Non-degenerate mixed functions, Kodai Math. J., Tome 33 (2010) no. 1, pp. 1-62 | Article | MR 2732230 | Zbl 1195.14061

[26] Parusiński, Adam On the bifurcation set of complex polynomial with isolated singularities at infinity, Compositio Math., Tome 97 (1995) no. 3, pp. 369-384 | Numdam | MR 1353280 | Zbl 0840.32007

[27] Pinchuk, Sergey A counterexample to the strong real Jacobian conjecture, Math. Z., Tome 217 (1994) no. 1, pp. 1-4 | Article | MR 1292168 | Zbl 0874.26008

[28] Rabier, Patrick J. Ehresmann fibrations and Palais-Smale conditions for morphisms of Finsler manifolds, Ann. of Math. (2), Tome 146 (1997) no. 3, pp. 647-691 | Article | MR 1491449 | Zbl 0919.58003

[29] Siersma, Dirk; Tibăr, Mihai Singularities at infinity and their vanishing cycles, Duke Math. J., Tome 80 (1995) no. 3, pp. 771-783 | Article | MR 1370115 | Zbl 0871.32024

[30] Suzuki, Masakazu Propriétés topologiques des polynômes de deux variables complexes, et automorphismes algébriques de l’espace C 2 , J. Math. Soc. Japan, Tome 26 (1974), pp. 241-257 | Article | MR 338423 | Zbl 0276.14001

[31] Tibăr, Mihai Regularity at infinity of real and complex polynomial functions, Singularity theory (Liverpool, 1996), Cambridge Univ. Press, Cambridge (London Math. Soc. Lecture Note Ser.) Tome 263 (1999), pp. xx, 249-264 | MR 1709356 | Zbl 0930.58005

[32] Tibăr, Mihai Polynomials and vanishing cycles, Cambridge University Press, Cambridge, Cambridge Tracts in Mathematics, Tome 170 (2007), pp. xii+253 | Article | MR 2360234 | Zbl 1126.32026

[33] Tibăr, Mihai; Zaharia, Alexandru Asymptotic behaviour of families of real curves, Manuscripta Math., Tome 99 (1999) no. 3, pp. 383-393 | Article | MR 1702581 | Zbl 0965.14012

[34] Verdier, Jean-Louis Stratifications de Whitney et théorème de Bertini-Sard, Invent. Math., Tome 36 (1976), pp. 295-312 | Article | MR 481096 | Zbl 0333.32010