From the sixteenth Hilbert problem to tropical geometry
type: Item
instance of: scholarly article
title: From the sixteenth Hilbert problem to tropical geometry
zbMATH Open document ID: 1185.14053
published in: Japanese Journal of Mathematics. 3rd Series
publication date: 2009-12-09T00:00:00Z
review text: This paper is a beautiful travel through time, from year 1900 to the present. It connects Hilbert's sixteenth problem (one of the few still open, after 110 years!) to tropical geometry. Having checked the original text by Hilbert, the paper's author claims that Hilbert's sixteenth problem was solved long ago! Let me explain this apparent contradiction. Hilbert's sixteenth is the ``Problem of the topology of algebraic curves and surfaces''. Hilbert's question originated from a result of Harnack, by which the maximum number of connected components of a real algebraic projective plane curve of degree \(d\) is \(\dfrac{(d-1)(d-2)}{2}+1\). Hilbert expressed the question in somewhat vague terms, asking for a ``thorough investigation of the relative position of the separate branches'' (i.e., the connected components of real algebraic curves in the plane) ``when their number is the maximum'' and ``the corresponding investigation as to the number, form and position of the sheets of an algebraic surface in space''. For the sake of concreteness, Hilbert mentioned plane sextic curves and quartic surfaces in 3-dimensional space. It was implicit that these curves and surfaces were defined over the reals. Using modern terms, one can rephrase Hilbert's problem as a classification problem (isotopy classification, or rigid isotopy classification, etc.) and it has little chances to be completed, as the author admits. The isotopy classification question was solved for sextic curves by \textit{D. A. Gudkov} [Mat. Sb., N. Ser. 67(109), 481--527 (1965; Zbl 0163.15502)], for curves of degree 7 by \textit{O. Viro} [Sov. Math. Dokl. 22, 566--570 (1980; Zbl 0481.14009)] and for quartic surfaces by \textit{V. M. Kharlamov} [Funct. Anal. Appl. 6, 345--346 (1972; Zbl 0276.14019); Funkts. Anal. Prilozh. 12, No. 1, 86--87 (1978; Zbl 0409.57031)] and \textit{V. V. Nikulin} [Math. USSR, Izv. 14, 103--167 (1980; Zbl 0408.10011)]. As we can see, big progress has been achieved by various mathematicians…
Mathematics Subject Classification ID: 14P25
Mathematics Subject Classification ID: 14N10
zbMATH DE Number: 5644702
zbMATH Keywords: sixteenth Hilbert problem
zbMATH Keywords: real algebraic curve
zbMATH Keywords: patchwork
zbMATH Keywords: tropical geometry
zbMATH Keywords: idempotent mathematics
author: Oleg Viro
reviewed by: María Jesús de la Puente
MaRDI profile type: Publication
full work available at URL: s11537-008-0832-6
OpenAlex ID: W1992224331
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DOI: 10.1007/S11537-008-0832-6
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description: scientific article; zbMATH DE number 5644702