From the sixteenth Hilbert problem to tropical geometry
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 from the Russian school (Viro, the paper's author, being a remarkable member of it) and a few from the French school. But, nevertheless, many questions remain open for curves and surfaces of arbitrary degree. Let me concentrate on curves. The problem is twofold. First, one must find as many relations as possible among the topological invariants of a curve of given degree (an example of this is the Gudkov--Rokhlin congruence). Only some arrangements of branches satisfy these restrictions. Now, one must construct examples of curves (i.e., give the coefficients of the defining polynomials), for those valid arrangements; in other words, one must provide examples which realize those arrangements. The restrictions arise from the topology of the complexification of the curve \(\mathcal{C}\), which is a closed oriented surface \(\mathcal{S}\) (this was already noticed by Klein). For instance, either the curve \(\mathcal{C}\) disconnects the surface \(\mathcal{S}\) into two connected components or \(\mathcal{S}\setminus \mathcal{C}\) is connected. Harnack, Hilbert, Rohn and Gudkov devised some methods to construct examples. These were good for \(d\leq6\) but insufficient for degree 7 curves. In 1979-1980 Viro presented a new technique of construction, called \textit{patchwork}, which allowed to settle the degree 7 case, as well as many instances for degree 8. Combinatorial patchwork is a simple kind of patchwork. From some combinatorial data, it provides a real polynomial \(f\) in two variables such that the branches of the curve \(f=0\) are arranged in a certain way (and this arrangement, in a piecewise--linear look, can be easily read from the data). In short and more generally, combinatorial patchwork is a way to obtain a real algebraic hypersurface (curve, surface, etc), from a PL object. Moreover, it is possible to go the other way round. The way back is called \textit{dequantization}. Given an algebraic hypersurface \(A\) (defined over the real or even the complex field), one can draw a version \(A_h\) of \(A\), on base \(h\) double--logarithmic paper (taking absolute values, first). Then, the limit of the \(A_h\)'s, when \(h\) tends to zero, is a PL object. This pass to the limit has been called \textit{dequantization}. And, in the limit, we have the so called \textit{tropical mathematics}. This means working over the reals, where \(\max\{a,b\}\) is conceived as the sum of \(a\) and \(b\), and \(a+b\) is conceived as the product of \(a\) and \(b\), for \(a,b,\in\mathbb{R}\). Tropical mathematics is also called \textit{idempotent mathematics}, since \(\max\{a,a\}=a\). Here, let me quote S. Banach: ``A mathematician is a person who can find analogies between theorems; a better mathematician is one who can see analogies between proofs and the best mathematician can notice analogies between theories. One can imagine that the ultimate mathematician is one who can see analogies between analogies.'' It turns out that considering the tropical (or idempotent) versions of various important topics in mathematics provides striking correspondences and analogies, as Litvinov, Maslov and others have remarked. This is why tropical mathematics seems worthy to me. Last, in page 199, the author gives a list of related open problems. One of them is, of course, the second part of Hilbert's sixteenth problem: ``the topology of families of curves defined by differential equations''.
scientific article; zbMATH DE number 5644702
scientific article