Neither real analytic sets nor the images of real or complex analytic mappings are, in general, coherent. Let be a morphism of real analytic spaces, and let be a homomorphism of coherent modules over the induced ring homomorphism . We conjecture that, despite the failure of coherence, certain natural discrete invariants of the modules of formal relations , , are upper semi-continuous in the analytic Zariski topology of . We prove semicontinuity in many cases (e.g. in the algebraic category). Semicontinuity of the “diagram of initial exponents” provides a unified point of view and explicit new techniques which substitute for coherence in both geometric problems on the images of mappings (semianalytic and subanalytic sets) and analytic problems on the singularities of differentiable functions (in particular, the classical division and composition problems).
Ni les ensembles analytiques réels, ni les images d’applications analytiques réelles ou complexes sont cohérents, en général. Soit un morphisme d’espaces analytiques, et soit un homomorphisme de modules cohérents au-dessus de l’homomorphisme induit d’anneaux . On conjecture que, malgré le manque de cohérence, certains invariants discrets naturels des modules de relations formelles , , sont semicontinus supérieurement pour la topologie de Zariski analytique de . On démontre la semicontinuité dans plusieurs cas (par exemple, dans la catégorie algébrique). La semicontinuité du “diagramme des exposants initiaux” fournit un point de vue unifié et des techniques nouvelles et explicites qui se substituent à la cohérence dans des problèmes géoémtriques sur les images d’applications (ensembles semianalytiques ou sousanalytiques) et dans des problèmes analytiques sur les singularités de fonctions différentiables (en particulier, les problèmes classiques de division et composition).
@article{AIF_1987__37_1_187_0, author = {Bierstone, Edward and Milman, P. D.}, title = {Relations among analytic functions. {I}}, journal = {Annales de l'Institut Fourier}, pages = {187--239}, publisher = {Institut Fourier}, address = {Grenoble}, volume = {37}, number = {1}, year = {1987}, doi = {10.5802/aif.1082}, zbl = {0611.32002}, mrnumber = {88g:32013a}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1082/} }
TY - JOUR AU - Bierstone, Edward AU - Milman, P. D. TI - Relations among analytic functions. I JO - Annales de l'Institut Fourier PY - 1987 SP - 187 EP - 239 VL - 37 IS - 1 PB - Institut Fourier PP - Grenoble UR - https://aif.centre-mersenne.org/articles/10.5802/aif.1082/ DO - 10.5802/aif.1082 LA - en ID - AIF_1987__37_1_187_0 ER -
%0 Journal Article %A Bierstone, Edward %A Milman, P. D. %T Relations among analytic functions. I %J Annales de l'Institut Fourier %D 1987 %P 187-239 %V 37 %N 1 %I Institut Fourier %C Grenoble %U https://aif.centre-mersenne.org/articles/10.5802/aif.1082/ %R 10.5802/aif.1082 %G en %F AIF_1987__37_1_187_0
Bierstone, Edward; Milman, P. D. Relations among analytic functions. I. Annales de l'Institut Fourier, Volume 37 (1987) no. 1, pp. 187-239. doi : 10.5802/aif.1082. https://aif.centre-mersenne.org/articles/10.5802/aif.1082/
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