Geodesic
On two invariants of divisorial valuations at infinity
Journal of Algebraic Combinatorics, Tome 39 (2014) no. 3, pp. 691-710.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

The two-dimensional case of the famous Jacobian conjecture of O.-H. Keller asserts that every unramified polynomial self-map of an affine plane is invertible. Many geometric approaches to this conjecture involve divisorial valuations of the field $\mathbb C(x,y)$, centered outside of the affine plane. Two integer invariants of these valuations naturally appear in this context. In this paper we study these invariants using combinatorics of weighted graphs. In particular, we prove that whenever both invariants are fixed, the corresponding valuations form a finite number of families up to plane automorphisms.
Classification : 14R15, 13F30, 05C05, 05C78, 14E05, 14J99
Keywords: divisorial valuation, weighted graph, blowup, Jacobian conjecture 
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     author = {Borisov, Alexander},
     title = {On two invariants of divisorial valuations at infinity},
     journal = {Journal of Algebraic Combinatorics},
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     number = {3},
     year = {2014},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JAC_2014__39_3_a3/}
}
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Borisov, Alexander. On two invariants of divisorial valuations at infinity. Journal of Algebraic Combinatorics, Tome 39 (2014) no. 3, pp. 691-710. http://geodesic.mathdoc.fr/item/JAC_2014__39_3_a3/