Geodesic
Analytic continuation of diagonals of Laurent series for rational functions
Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 14 (2021) no. 3, pp. 360-368.

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We describe branch points of complete $\boldsymbol{q}$-diagonals of Laurent series for rational functions in several complex variables in terms of the logarithmic Gauss mapping. The sufficient condition of non-algebraicity of such a diagonal is proven.
Keywords: diagonals of Laurent series, hyperplane amoeba, logarithmic Gauss mapping, zero pinch
Mots-clés : monodromy. 
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Dmitry Yu. Pochekutov. Analytic continuation of diagonals of Laurent series for rational functions. Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 14 (2021) no. 3, pp. 360-368. http://geodesic.mathdoc.fr/item/JSFU_2021_14_3_a8/

[1] I. Gelfand, M. Kapranov, A. Zelevinsky, Discriminants, resultants and multidimensional determinants, Birkhäuser, 1994

[2] R. Stanley, Enumerative combinatorics, v. 2, Cambridge University Press, 1999

[3] D. Pochekutov, “Diagonals of the Laurent Series of Rational Functions”, Siberian Math. Journal, 50:6 (2009), 1081–1091 | DOI

[4] H. Furstenberg, “Algebraic functions over finite fields”, Journal of Algebra, 7:2 (1967), 271–277

[5] H. Poincaré, Les méthodes nouvelles de la mécanique céleste, v. I, Gauthier-Villars et fils, Paris, 1892

[6] G. Pólya, “Sur les séries entières, dont la somme est une fonction algébrique”, Enseign. Math., 22 (1921), 38–47

[7] M.L.J. Hautus, D.A. Klarner, “The diagonal of a double power series”, Duke Math. J., 38:2 (1971), 229–235

[8] D. Ž.Djokovič, “A properties of the Taylor expansion of rational function in several variables”, J. Math. Anal. Appl., 66 (1978), 679–685

[9] M. Forsberg, M. Passare, A. Tsikh, “Laurent Determinants and Arrangements of Hyperplane Amoebas”, Advances in Mathematics, 151:1 (2000), 45–70

[10] D. Fotiadi, M. Froissart, J. Lascoux, F. Pham, “Applications of an isotopy theorem”, Topology, 4:2 (1965), 159–191

[11] C.W. Curtis, I. Reiner, Representation Theory of Finite Groups and Associative Algebras, Interscience Publishers, 1962

[12] V.A. Vasiliev, Applied Picard-Lefschetz Theory, AMS, 2002

[13] E.W. Barnes, “A new development of the theory of the hypergeometric functions”, Proc. Lond. Math. Soc. (2), 6:1 (1908), 141–177

[14] T. Clausen, “Ueber die Fälle, wenn die Reihe von der Form $y = 1 + \ldots$ etc. ein Quadrat von der Form $z = 1 +\ldots$ etc. hat”, Journal für die reine und angewandte Mathematik, 3 (1828), 89–91