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Uspekhi Mat. Nauk, 2004, Volume 59, Issue 4(358), Pages 69–146 (Mi umn759)  

This article is cited in 9 scientific papers (total in 9 papers)

On solvability and unsolvability of equations in explicit form

A. G. Khovanskii

Independent University of Moscow

Abstract: In this survey the classical results of Abel, Liouville, Galois, Picard, Vessiot, Kolchin, and others on the solvability and unsolvability of equations in explicit form are discussed. The one-dimensional topological version of Galois theory is presented in detail (this version describes topological obstructions to the representability of functions by quadratures).

DOI: https://doi.org/10.4213/rm759

Full text: PDF file (757 kB)
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English version:
Russian Mathematical Surveys, 2004, 59:4, 661–736

Bibliographic databases:

UDC: 512.623.3+517.547.59
MSC: Primary 12F10, 12H05, 34M35; Secondary 32S40, 34M50, 30F99, 32D15, 20F16
Received: 30.03.2004

Citation: A. G. Khovanskii, “On solvability and unsolvability of equations in explicit form”, Uspekhi Mat. Nauk, 59:4(358) (2004), 69–146; Russian Math. Surveys, 59:4 (2004), 661–736

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. A. G. Khovanskii, “Variations on the Theme of Solvability by Radicals”, Proc. Steklov Inst. Math., 259 (2007), 82–100  mathnet  crossref  mathscinet  zmath  elib  elib
    2. Yu. G. Kudryashov, “Transfer of complex structures and topological character of holomorphy”, J. Math. Sci. (N. Y.), 161:3 (2009), 388–391  mathnet  crossref
    3. V. Ya. Lin, “Algebraic functions, configuration spaces, Teichmüller spaces, and new holomorphically combinatorial invariants”, Funct. Anal. Appl., 45:3 (2011), 204–224  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    4. Alexandersson P., “On Eigenvalues of the Schrodinger Operator with an Even Complex-Valued Polynomial Potential”, Comput. Methods Funct. Theory, 12:2 (2012), 465–481  crossref  zmath  isi  elib
    5. Alexandersson P., Gabrielov A., “On Eigenvalues of the Schrodinger Operator with a Complex-Valued Polynomial Potential”, Comput. Methods Funct. Theory, 12:1 (2012), 119–144  crossref  mathscinet  zmath  isi
    6. S. A. Stepin, “Non-integrability by quadratures and formal asymptotic solutions of the Riccati equation”, Russian Math. Surveys, 69:6 (2014), 1123–1125  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    7. Kuleshov A.S., Dobrynin D.S., “Motion of a Rotationally Symmetric Ellipsoid on a Fixed Perfectly Rough Horizontal Plane”, 2015 International Conference on Mechanics Seventh Polyakhovs Reading, ed. Tikhonov A., IEEE, 2015  isi
    8. Dobrynin D.S., Kuleshov A.S., “Solvable Cases in the Problem of Motion of a Heavy Rotationally Symmetric Ellipsoid on a Perfectly Rough Plane”, Iutam Symposium Analytical Methods in Nonlinear Dynamics, Procedia Iutam, 19, eds. Hagedorn P., Clerkin E., Elsevier Science BV, 2016, 161–168  crossref  isi
    9. Moulay A. Barkatou, Renat R. Gontsov, “Linear Differential Systems with Small Coefficients: Various Types of Solvability and their Verification”, SIGMA, 15 (2019), 058, 15 pp.  mathnet  crossref
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