Hilbert problem 15 and Ritt-Wu method. I
type: Item
instance of: scholarly article
title: Hilbert problem 15 and Ritt-Wu method. I
zbMATH Open document ID: 1411.14061
published in: Journal of Systems Science and Complexity
publication date: 2019-03-06T00:00:00Z
review text: The article under review provides a proof of a theorem in H. Schubert's classical text [Calcül der abzählenden Geometrie. Leipzig: Teubner (1879; JFM 11.0460.01)] on the enumerative relations among six special lines associated to a given planar cuspidal cubic curve and a given point in the plane. A cuspidal cubic curve in the projective complex plane has three special points: its cusp point, its inflection point, and the intersection point of its inflection tangent line with the tangent line at its cusp. Given such a curve and a general point in the plane, there is six special lines passing through the point, namely the three lines which connect the given point with the three special points plus the three lines which are tangent to the curve and pass through the given point. Schubert's theorem states that, fixing any four of these six special lines, there are only finitely many possibilities for the other two. This theorem also gives the exact numbers of possibilities, depending on which four of the six lines have been fixed. To rephrase the theorem, one can consider the map which, for a fixed point in the plane, sends a planar cuspidal cubic curve to the six special lines associated to the curve and the fixed point. Its image lives in a product of six projective lines, each of these six factors consisting of all lines in the plane that pass through the fixed point. Schubert's theorem says that this image is four-dimensional and it states its multidegree. As Schubert did not provide a full formal proof of his theorem, the author of the article at hand shows that Schubert's theorem is correct. The proof combines theoretical insights with computations and makes heavy use of a Maple implementation of the Ritt-Wu method. In addition, the author proves that the singular locus of the four-dimensional image of the map described above is an irreducible surface.
reviewed by: Kathlén Kohn
Mathematics Subject Classification ID: 14N15
Mathematics Subject Classification ID: 14H45
Mathematics Subject Classification ID: 14Q05
zbMATH DE Number: 7032427
zbMATH Keywords: cuspidal cubic curves
zbMATH Keywords: Schubert calculus
zbMATH Keywords: multidegree
zbMATH Keywords: Maple computations
zbMATH Keywords: singular locus
zbMATH Keywords: Ritt-Wu method
author: Bang-He Li
describes a project that uses: Maple
MaRDI profile type: Publication
full work available at URL: s11424-019-8344-4
OpenAlex ID: W4229494772
cites work: Q3207678
cites work: Q5680234
cites work: Q5553838
cites work: Zero decomposition algorithms for systems of polynomial equations
cites work: An algorithm for constructing Gröbner bases from characteristic sets and its application to geometry
cites work: Irreducible decomposition of algebraic varieties via characteristic sets and Gröbner bases
DOI: 10.1007/S11424-019-8344-4
description: scientific article; zbMATH DE number 7032427