Extremal values of the basic invariants of plane curves (Q2706866)

The aim of this paper is to study plane curves. Here, a plane curve is a generic immersion of the oriented circle into the oriented Euclidean plane. Generic means that the curve has only ordinary double points at which the curve is transverse to itself. Denote by \(C(n,i)\) the set of all plane curves with \(n\) double points and Whitney index \(i\).NEWLINENEWLINENEWLINEThe authors study the three basic invariants \(J^+\), \(J^-\) and \(St\) of plane curves introduced by \textit{V. I. Arnold} [Adv. Sov. Math. 21, 33-91 (1994; Zbl 0864.57027)]. More precisely, they determine on which curves of \(C(n,i)\) the extremal values of \(J^+\), \(J^-\) and \(St\) are attained. This proves some conjectures concerning the extremal values of these invariants on the set \(C(n,i)\). For similar results see \textit{A. N.~Shumakovich} [St. Petersbg. Math. J. 7, No. 3, 445-472 (1996); translation from Algebra Anal. 7, No. 3, 165-199; Corrections No. 5, 252-254 (1995; Zbl 0890.57044)] and \textit{O. Viro} [Transl., Ser. 2, Am. Math. Soc. 173, 231-252 (1996; Zbl 0879.14031)].