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.
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.
Keywords: real and complex polynomial mappings, bifurcation locus, Jacobian problem, Newton polyhedron, regularity at infinity
Mot 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{a}r, Mihai}, title = {Invertible polynomial mappings via {Newton} non-degeneracy}, journal = {Annales de l'Institut Fourier}, pages = {1807--1822}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {64}, number = {5}, year = {2014}, doi = {10.5802/aif.2897}, zbl = {06387324}, mrnumber = {3330924}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/aif.2897/} }
TY - JOUR AU - Chen, Ying AU - Dias, Luis Renato G. AU - Takeuchi, Kiyoshi AU - Tibăr, Mihai TI - Invertible polynomial mappings via Newton non-degeneracy JO - Annales de l'Institut Fourier PY - 2014 SP - 1807 EP - 1822 VL - 64 IS - 5 PB - Association des Annales de l’institut Fourier UR - http://archive.numdam.org/articles/10.5802/aif.2897/ DO - 10.5802/aif.2897 LA - en ID - AIF_2014__64_5_1807_0 ER -
%0 Journal Article %A Chen, Ying %A Dias, Luis Renato G. %A Takeuchi, Kiyoshi %A Tibăr, Mihai %T Invertible polynomial mappings via Newton non-degeneracy %J Annales de l'Institut Fourier %D 2014 %P 1807-1822 %V 64 %N 5 %I Association des Annales de l’institut Fourier %U http://archive.numdam.org/articles/10.5802/aif.2897/ %R 10.5802/aif.2897 %G en %F 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, Volume 64 (2014) no. 5, pp. 1807-1822. doi : 10.5802/aif.2897. http://archive.numdam.org/articles/10.5802/aif.2897/
[1] Injective morphisms of real algebraic varieties, Proc. Amer. Math. Soc., Volume 13 (1962), pp. 200-203 | DOI | MR | Zbl
[2] Injectivity of real polynomial maps and Łojasiewicz exponents at infinity, Math. Z., Volume 257 (2007) no. 4, pp. 745-767 | DOI | MR | Zbl
[3] On the topology of polynomial hypersurfaces, Singularities, Part 1 (Arcata, Calif., 1981) (Proc. Sympos. Pure Math.), Volume 40, Amer. Math. Soc., Providence, RI, 1983, pp. 167-178 | MR | Zbl
[4] Milnor numbers and the topology of polynomial hypersurfaces, Invent. Math., Volume 92 (1988) no. 2, pp. 217-241 | DOI | MR | Zbl
[5] On Newton non-degeneracy of polynomial mappings (arXiv:1207.1612)
[6] Bifurcation values and monodromy of mixed polynomials, Math. Res. Lett., Volume 19 (2012) no. 1, pp. 59-79 | DOI | MR | Zbl
[7] Injective endomorphisms of algebraic and analytic sets, Ann. Polon. Math., Volume 56 (1991) no. 1, pp. 29-35 | MR | Zbl
[8] Regularity at infinity of real mappings and a Morse-Sard theorem, J. Topol., Volume 5 (2012) no. 2, pp. 323-340 | DOI | MR | Zbl
[9] Five definitions of critical point at infinity, Singularities (Oberwolfach, 1996) (Progr. Math.), Volume 162, Birkhäuser, Basel, 1998, pp. 345-360 | MR | Zbl
[10] Polynomial automorphisms and the Jacobian conjecture, Progress in Mathematics, 190, Birkhäuser Verlag, Basel, 2000, pp. xviii+329 | DOI | Zbl
[11] Motivic Milnor fibers over complete intersection varieties and their virtual Betti numbers, Int. Math. Res. Not. IMRN (2012) no. 15, pp. 3567-3613 | DOI | MR | Zbl
[12] Fibers of polynomial mappings at infinity and a generalized Malgrange condition, Compositio Math., Volume 119 (1999) no. 2, pp. 157-167 | DOI | MR | Zbl
[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 | EuDML | Numdam | Zbl
[14] Sur la topologie des polynômes complexes, Acta Math. Vietnam, Volume 9 (1984) no. 1, pp. 21-32 | Zbl
[15] Testing sets for properness of polynomial mappings, Math. Ann., Volume 315 (1999) no. 1, pp. 1-35 | DOI | MR | Zbl
[16] On asymptotic critical values and the Rabier theorem, Geometric singularity theory (Banach Center Publ.), Volume 65, Polish Acad. Sci., Warsaw, 2004, pp. 125-133 | DOI | MR | Zbl
[17] Semialgebraic Sard theorem for generalized critical values, J. Differential Geom., Volume 56 (2000) no. 1, pp. 67-92 http://projecteuclid.org/getRecord?id=euclid.jdg/1090347525 | MR | Zbl
[18] Polyèdres de Newton et nombres de Milnor, Invent. Math., Volume 32 (1976), pp. 1-31 | DOI | EuDML | Zbl
[19] Monodromy zeta functions at infinity, Newton polyhedra and constructible sheaves, Math. Z., Volume 268 (2011) no. 1-2, pp. 409-439 | DOI | MR | Zbl
[20] On the bifurcation set of a polynomial function and Newton boundary, Publ. Res. Inst. Math. Sci., Volume 26 (1990) no. 4, pp. 681-689 | DOI | MR | Zbl
[21] Milnor fibration at infinity, Indag. Math. (N.S.), Volume 3 (1992) no. 3, pp. 323-335 | DOI | MR | Zbl
[22] Bifurcation set, -tameness, asymptotic critical values and Newton polyhedrons, Kodai Math. J., Volume 36 (2013) no. 1, pp. 77-90 | DOI | MR | Zbl
[23] Non-degenerate complete intersection singularity, Actualités Mathématiques. [Current Mathematical Topics], Hermann, Paris, 1997, pp. viii+309 | MR | Zbl
[24] Topology of polar weighted homogeneous hypersurfaces, Kodai Math. J., Volume 31 (2008) no. 2, pp. 163-182 | DOI | MR | Zbl
[25] Non-degenerate mixed functions, Kodai Math. J., Volume 33 (2010) no. 1, pp. 1-62 | DOI | MR | Zbl
[26] On the bifurcation set of complex polynomial with isolated singularities at infinity, Compositio Math., Volume 97 (1995) no. 3, pp. 369-384 | EuDML | Numdam | MR | Zbl
[27] A counterexample to the strong real Jacobian conjecture, Math. Z., Volume 217 (1994) no. 1, pp. 1-4 | DOI | EuDML | MR | Zbl
[28] Ehresmann fibrations and Palais-Smale conditions for morphisms of Finsler manifolds, Ann. of Math. (2), Volume 146 (1997) no. 3, pp. 647-691 | DOI | MR | Zbl
[29] Singularities at infinity and their vanishing cycles, Duke Math. J., Volume 80 (1995) no. 3, pp. 771-783 | DOI | MR | Zbl
[30] Propriétés topologiques des polynômes de deux variables complexes, et automorphismes algébriques de l’espace , J. Math. Soc. Japan, Volume 26 (1974), pp. 241-257 | DOI | MR | Zbl
[31] Regularity at infinity of real and complex polynomial functions, Singularity theory (Liverpool, 1996) (London Math. Soc. Lecture Note Ser.), Volume 263, Cambridge Univ. Press, Cambridge, 1999, pp. xx, 249-264 | MR | Zbl
[32] Polynomials and vanishing cycles, Cambridge Tracts in Mathematics, 170, Cambridge University Press, Cambridge, 2007, pp. xii+253 | DOI | MR | Zbl
[33] Asymptotic behaviour of families of real curves, Manuscripta Math., Volume 99 (1999) no. 3, pp. 383-393 | DOI | MR | Zbl
[34] Stratifications de Whitney et théorème de Bertini-Sard, Invent. Math., Volume 36 (1976), pp. 295-312 | DOI | EuDML | MR | Zbl
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