Systematic Enumeration of Formally Admissible Real Schemes of Nonsingular Plane Algebraic Curves of Degree 8

Degree 8 is the first degree for which a complete classification of real schemes of nonsingular plane algebraic curves remains unresolved. Even a complete list of formally admissible schemes satisfying known necessary conditions is absent from the literature. In the present work, a systematic enumeration of all formally ad missible real schemes for degree 8 is carried out using restrictions 2.2.A–2.2.F from Viro’s work [4]. A total of 2926 formally admissible schemes are obtained over all numbers of ovals N = 0, . . . , 22, including 104 schemes for M-curves (N = 22), 243 for (M−1)-curves (N = 21), and 409 for (M−2)-curves (N = 20). The re sult for M-curves coincides with the known value from the works of Viro [4] and Orevkov [6]. The results for N = 20 and N = 19 coincide with Viro’s survey [12]. A comparison with the lower bound on the number of T-curves, ≥ 2367, reveals a gap of ≤ 559 schemes. For each of the seven structural types, explicit formulas are derived expressing the number of schemes as a function of the number of ovals N.