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On holonomic systems for $$\prod_{l=1}^N(f_l+\sqrt{-1}0)^{\lambda_l}$$. (English) Zbl 0449.35067

##### MSC:
 35N10 Overdetermined systems of PDEs with variable coefficients 35G05 Linear higher-order PDEs 35A99 General topics in partial differential equations 46F15 Hyperfunctions, analytic functionals 14E15 Global theory and resolution of singularities (algebro-geometric aspects)
##### Keywords:
holonomic systems; characteristic variety; hyperfunction
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##### References:
 [1] Aroca, J. M., H. Hironaka and J. L. Vicente, The Theory of the Maximal Contact. Memorias de Matematica del Institute Jorge Juan, 29. Madrid: Consejo Superior de Investigaciones Cientificas, 1975. · Zbl 0366.32008 [2] Atiyah, M. F., Resolution of singularities and division of distributions, Comm. Pure Appl. Math., 23 (1970), 145-150. · Zbl 0188.19405 · doi:10.1002/cpa.3160230202 [3] Bernstein, I. N., Modules over a ring of differential operators, Study of the funda- mental solutions of equations with constants, Funkcional. Anal, i Prilozen, 5 (1970), 1-16(in Russian). · Zbl 0233.47031 · doi:10.1007/BF01076413 [4] Bernstein, L N. and S. L Gel’fand, Meromorphic properties of the function P Funkcional. Anal, i Prilozen, 3 (1969), 84-85 (in Russian). · Zbl 0208.15201 · doi:10.1007/BF01078276 [5] Hironaka, H., Introduction to the theory of infinitely near singular points, Memo- rias de Matematica del Institute Jorge Juan, 28, Madrid: Consejo Superior de Investigaciones Cientificas, 1974. · Zbl 0366.32007 [6] Kashiwara, M., J3-functions and holonomic systems, Invent. Math., 38 (1976), 33-53. · Zbl 0354.35082 [7] Kashiwara, M. and T. Kawai, Micro-local properties of II f j i , Proc. Japan Acad., 51 (1975), 270-272. · Zbl 0304.35068 · doi:10.3792/pja/1195518633 [8] , On a conjecture of Regge and Sato on Feynman integrals, Proc. Japan Acad., 52 (1976), 161-164. · Zbl 0347.35061 · doi:10.3792/pja/1195518341 [9] Kashiwara, M., T. Kawai and H. P. Stapp, Micro-analytic structure of the S-matrix and related functions, Publ. RIMS, Kyoto Univ. 12 Supp., (1977), 141-146. A full paper titled ” Micro-analyticity of the S-matrix and related functions” will appear in Commun. math. Phys., 66 (1979). · Zbl 0443.35059 · doi:10.2977/prims/1195196603 [10] Sato, M., T. Kawai and M. Kashiwara, Microfunctions and pseudo-differential equa- tions, Lecture Notes in Math., 287, 263-529. Berlin-Heidelberg-New York, Springer, 1973. · Zbl 0277.46039 [11] Whitney, H., Tangents to an analytic variety, Ann. of Math., 81 (1964), 496-549. · Zbl 0152.27701 · doi:10.2307/1970400 [12] lagolnitzer, D., The u=Q structure theorem, Comm. math. Phys., 63 (1978), 49-96. Added in proof: The statement in page 9, line 4 is erroneous, because P/s and Qj’s do not satisfy the commutation relation. Hence the proof of Proposition 7 is not correct. Although the proof of Theorem 2 depends on Proposition 7, if we replace JTjt9 with -^O.pO^, £), it does not depend on Proposition 7. Here a= (ai, ●●●, ai), /9= (/?i, ●●●, &) eCl and Jfjt<f(ct,$$) is obtained from J^s.9 by letting sj subject to the relation Sj-ajS+fij with one indeterminate s. Theorfore the proof of Theorem 18 is complete as it stands. The detailed corrections will be submitted to this journal. See also our paper "On the charac- teristic variety of a holonomic system with regular singularities," which will appear in Adv. in Math. It gives a complete proof for a generalization of Theorem 18.$$
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