Sextactic points on a simple closed curve. (English) Zbl 1088.53049
Let \(\gamma\) be a regular curve in the real or complex projective plane. A point \(p \in \gamma\) is said to be sextatic if the osculatory conic meets \(\gamma\) with multiplicity at least six at \(p\). Like inflexion points, sextatic points of curves in the complex projective plane are well understood since the nineteenth century, but the case of (not necessarily algebraic) curves in the real projective plane \(P^2\) still poses problems. Mukhopadhyaya proved in 1909 that strictly convex curves in the affine plane have at least six sextatic points. However, apparently nothing was known about curves that are not strictly convex.
Let \(\gamma\) be a simple, closed, regular, \(C^{\infty}\)-parameterized curve in \(P^2\). The main results the authors prove are summarized in the following three theorems.
1) If \(\gamma\) is not nullhomotopic, then it has at least three sextatic points.
2) If \(\gamma\) is nullhomotopic, then it has at least two or three sextatic points according to whether it is convex or not.
3) If \(\gamma\) is nullhomotopic, then the total number of sextatic and inflection points of \(\gamma\) is at least four.
In particular, 1) answers positively a long standing conjecture of G. Bol. Moreover, the authors show that all these results, as well as the projective version of Mukhopadhyaya’s, are optimal. They also observe that all theorems remain true for \(C^4\)-curves, up to modify the notion of sextatic point appropriately. The authors’ approach is axiomatic and similar to the one they developed to deal with vertices [A unified approach to the four vertex theorems, I, II, Differential and symplectic topology of knots and curves S. Tabachnikov (ed.) Am. Math. Soc. Transl., Ser. 2, 190(42), 185–228 (1999; Zbl 1068.53005) and ibid. 190(42), 229–252 (1999; Zbl 1068.53004)]. In the final appendix the authors consider curves in the complex projective plane and prove a formula counting the number of sextatic points with multiplicities for a nonsingular algebraic curve of degree \(d\). This is done without assuming that the inflection points are simple, as implicitly Cayley did finding the number \(3d(4d-9)\), in 1865.
Let \(\gamma\) be a simple, closed, regular, \(C^{\infty}\)-parameterized curve in \(P^2\). The main results the authors prove are summarized in the following three theorems.
1) If \(\gamma\) is not nullhomotopic, then it has at least three sextatic points.
2) If \(\gamma\) is nullhomotopic, then it has at least two or three sextatic points according to whether it is convex or not.
3) If \(\gamma\) is nullhomotopic, then the total number of sextatic and inflection points of \(\gamma\) is at least four.
In particular, 1) answers positively a long standing conjecture of G. Bol. Moreover, the authors show that all these results, as well as the projective version of Mukhopadhyaya’s, are optimal. They also observe that all theorems remain true for \(C^4\)-curves, up to modify the notion of sextatic point appropriately. The authors’ approach is axiomatic and similar to the one they developed to deal with vertices [A unified approach to the four vertex theorems, I, II, Differential and symplectic topology of knots and curves S. Tabachnikov (ed.) Am. Math. Soc. Transl., Ser. 2, 190(42), 185–228 (1999; Zbl 1068.53005) and ibid. 190(42), 229–252 (1999; Zbl 1068.53004)]. In the final appendix the authors consider curves in the complex projective plane and prove a formula counting the number of sextatic points with multiplicities for a nonsingular algebraic curve of degree \(d\). This is done without assuming that the inflection points are simple, as implicitly Cayley did finding the number \(3d(4d-9)\), in 1865.
Reviewer: Antonio Lanteri (Milano)
MSC:
| 53C75 | Geometric orders, order geometry |
| 51L15 | \(n\)-vertex theorems via direct methods |
| 53A20 | Projective differential geometry |
| 58K15 | Topological properties of mappings on manifolds |
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