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Tropical analytic geometry, Newton polygons, and tropical intersections

Publication ,  Journal Article
Rabinoff, J
Published in: Advances in Mathematics
April 1, 2012

In this paper we use the connections between tropical algebraic geometry and rigid-analytic geometry in order to prove two main results. We use tropical methods to prove a theorem about the Newton polygon for convergent power series in several variables: if f 1, ..., f n are n convergent power series in n variables with coefficients in a non-Archimedean field K, we give a formula for the valuations and multiplicities of the common zeros of f 1, ..., f n. We use rigid-analytic methods to show that stable complete intersections of tropical hypersurfaces compute algebraic multiplicities even when the intersection is not tropically proper. These results are naturally formulated and proved using the theory of tropicalizations of rigid-analytic spaces, as introduced by Einsiedler, Kapranov, and Lind (2006) [14] and Gubler (2007) [20]. We have written this paper to be as readable as possible both to tropical and arithmetic geometers. © 2012 Elsevier Inc..

Duke Scholars

Published In

Advances in Mathematics

DOI

EISSN

1090-2082

ISSN

0001-8708

Publication Date

April 1, 2012

Volume

229

Issue

6

Start / End Page

3192 / 3255

Related Subject Headings

  • General Mathematics
  • 4904 Pure mathematics
  • 4902 Mathematical physics
  • 4901 Applied mathematics
  • 0101 Pure Mathematics
 

Citation

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MLA
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Rabinoff, J. (2012). Tropical analytic geometry, Newton polygons, and tropical intersections. Advances in Mathematics, 229(6), 3192–3255. https://doi.org/10.1016/j.aim.2012.02.003
Rabinoff, J. “Tropical analytic geometry, Newton polygons, and tropical intersections.” Advances in Mathematics 229, no. 6 (April 1, 2012): 3192–3255. https://doi.org/10.1016/j.aim.2012.02.003.
Rabinoff J. Tropical analytic geometry, Newton polygons, and tropical intersections. Advances in Mathematics. 2012 Apr 1;229(6):3192–255.
Rabinoff, J. “Tropical analytic geometry, Newton polygons, and tropical intersections.” Advances in Mathematics, vol. 229, no. 6, Apr. 2012, pp. 3192–255. Scopus, doi:10.1016/j.aim.2012.02.003.
Rabinoff J. Tropical analytic geometry, Newton polygons, and tropical intersections. Advances in Mathematics. 2012 Apr 1;229(6):3192–3255.
Journal cover image

Published In

Advances in Mathematics

DOI

EISSN

1090-2082

ISSN

0001-8708

Publication Date

April 1, 2012

Volume

229

Issue

6

Start / End Page

3192 / 3255

Related Subject Headings

  • General Mathematics
  • 4904 Pure mathematics
  • 4902 Mathematical physics
  • 4901 Applied mathematics
  • 0101 Pure Mathematics