Dlog and formal flow for analytic isomorphisms of nspace
Authors:
David Wright and Wenhua Zhao
Journal:
Trans. Amer. Math. Soc. 355 (2003), 31173141
MSC (2000):
Primary 14R10, 11B68; Secondary 14R15, 05C05
Published electronically:
April 14, 2003
MathSciNet review:
1974678
Fulltext PDF Free Access
Abstract 
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Similar Articles 
Additional Information
Abstract: Given a formal map of the form terms, we give tree expansion formulas and associated algorithms for the DLog of and the formal flow . The coefficients that appear in these formulas can be viewed as certain generalizations of the Bernoulli numbers and the Bernoulli polynomials. Moreover, the coefficient polynomials in the formal flow formula coincide with the strict order polynomials in combinatorics for the partially ordered sets induced by trees. Applications of these formulas to the Jacobian Conjecture are discussed.
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 S. S. Abhyankar, Lectures in algebraic geometry, Notes by Chris Christensen, Purdue Univ., 1974.
 [BCW]
 H. Bass, E. Connell, and D. Wright, The Jacobian conjecture, reduction of degree and formal expansion of the inverse. Bull. Amer. Math. Soc. 7, (1982), 287330. MR 83k:14028. Zbl.539.13012.
 [CMTWW]
 C. C.A. Cheng, J. H. McKay, J. Towber, S. S.S. Wang, and D. Wright, Reversion of power series and the extended Raney coefficients, Trans. Amer. Math. Soc. 349 (1997), 17691782. MR 97h:13018 Zbl.868.13019.
 [Ge]
 I. M. Gessel, A combinatorial proof of the multivariable Lagrange inversion formula, J. Combin. Theory Ser. A, 45 (1987), 178195. MR 88h:05011 Zbl.651.05009.
 [Go]
 I. J. Good, Generalizations to several variables of Lagrange's expansion, with applications to stochastic processes, Proc. Cambridge Philos. Soc. 56 (1960), 367380. MR 23:A352 Zbl.135.18802.
 [HS]
 M. Haiman and W. Schmitt, Incidence algebra antipodes and Lagrange inversion in one and several variables, J. Combin. Theory Ser. A, 50 (1989), 172185. MR 90f:05005 Zbl.747.05007.
 [Ja]
 C. G. J. Jacobi, De resolutione aequationum per series infinitas, J. Reine Angew. Math. 184 (1830), 257286.
 [Jo]
 S. A. Joni, Lagrange inversion in higher dimensions and umbral operators, Linear and Multilinear Algebra 6 (1978) 111122. MR 58:10485 Zbl.395.05005.
 [R]
 G. N. Raney, Functional composition patterns and power series reversion, Trans. Amer. Math. Soc. 94 (1960), 441451. MR 22:5584 Zbl.131,14.
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 J. Shareshian, Personal communication.
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 J. Shareshian, D. Wright and W. Zhao, A New Realization of Order Polynomials. In Preparation.
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 W. Zhao, Exponential formulas for the Jacobians and Jacobian matrices of analytic maps, J. Pure Appl. Algebra, 166 (2002) 321336. MR 2002i:14059
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Additional Information
David Wright
Affiliation:
Department of Mathematics, Washington University in St. Louis, St. Louis, Missouri 631304899
Email:
wright@einstein.wustl.edu
Wenhua Zhao
Affiliation:
Department of Mathematics, Washington University in St. Louis, St. Louis, Missouri 631304899
Email:
zhao@math.wustl.edu
DOI:
http://dx.doi.org/10.1090/S0002994703032951
PII:
S 00029947(03)032951
Keywords:
Dlog,
formal flow of automorphisms,
rooted trees,
order polynomials,
Bernoulli numbers and polynomials
Received by editor(s):
June 5, 2002
Received by editor(s) in revised form:
January 3, 2003
Published electronically:
April 14, 2003
Article copyright:
© Copyright 2003
American Mathematical Society
