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.
@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] Injective morphisms of real algebraic varieties, Proc. Amer. Math. Soc., Tome 13 (1962), pp. 200-203 | Article | MR 140516 | Zbl 0107.14602
[2] 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] 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] 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] On Newton non-degeneracy of polynomial mappings (arXiv:1207.1612)
[6] 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] Injective endomorphisms of algebraic and analytic sets, Ann. Polon. Math., Tome 56 (1991) no. 1, pp. 29-35 | MR 1145567 | Zbl 0761.14005
[8] 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] 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] Polynomial automorphisms and the Jacobian conjecture, Birkhäuser Verlag, Basel, Progress in Mathematics, Tome 190 (2000), pp. xviii+329 | Article | Zbl 0962.14037
[11] 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] 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] 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] Sur la topologie des polynômes complexes, Acta Math. Vietnam, Tome 9 (1984) no. 1, pp. 21-32 | Zbl 0597.32005
[15] Testing sets for properness of polynomial mappings, Math. Ann., Tome 315 (1999) no. 1, pp. 1-35 | Article | MR 1717542 | Zbl 0946.14039
[16] 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] 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] Polyèdres de Newton et nombres de Milnor, Invent. Math., Tome 32 (1976), pp. 1-31 | Article | Zbl 0328.32007
[19] 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] 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] Milnor fibration at infinity, Indag. Math. (N.S.), Tome 3 (1992) no. 3, pp. 323-335 | Article | MR 1186741 | Zbl 0806.57021
[22] Bifurcation set, -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] Non-degenerate complete intersection singularity, Hermann, Paris, Actualités Mathématiques. [Current Mathematical Topics] (1997), pp. viii+309 | MR 1483897 | Zbl 0930.14034
[24] Topology of polar weighted homogeneous hypersurfaces, Kodai Math. J., Tome 31 (2008) no. 2, pp. 163-182 | Article | MR 2435890 | Zbl 1149.14031
[25] Non-degenerate mixed functions, Kodai Math. J., Tome 33 (2010) no. 1, pp. 1-62 | Article | MR 2732230 | Zbl 1195.14061
[26] 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] 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] 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] 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] Propriétés topologiques des polynômes de deux variables complexes, et automorphismes algébriques de l’espace , J. Math. Soc. Japan, Tome 26 (1974), pp. 241-257 | Article | MR 338423 | Zbl 0276.14001
[31] 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] 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] Asymptotic behaviour of families of real curves, Manuscripta Math., Tome 99 (1999) no. 3, pp. 383-393 | Article | MR 1702581 | Zbl 0965.14012
[34] Stratifications de Whitney et théorème de Bertini-Sard, Invent. Math., Tome 36 (1976), pp. 295-312 | Article | MR 481096 | Zbl 0333.32010