Newtonian and affine classifications of irreducible cubics (Q2849200)

Recall that an affine planar algebraic curve can be regarded as a curve in the projective plane \(\mathbb P^2\) equipped with a distinguished line, called line at infinity. Recall also that two real affine cubics \((C',L)\) and \((C'',L)\) are said to belong to the same \textit{Newtonian class} if they can be connected by an ambient isotopy \((\mathbb P^2_{\mathbb R},C_t,L_t)\) in the class of pairs \((\text{cubic},\text{line})\) during which singular/regular points of~\(C_t\) remain singular or regular, respectively, \textit{and the points of inflection of~\(C_t\) remain points of inflection}. In other words, a Newtonian isotopy is a rigid isotopy taking into account the inflection points. The author reconsiders (in somewhat more modern terms) Newton's classification of affine cubics, confining himself to the irreducible ones only. For each of the 59 Newtonian classes, he computes the modality (number of parameters up to projective equivalence) and the dimension of the corresponding stratum. In addition, explicit equations (with the correct number of parameters) are provided for each class.