Complex orientation formulas for M-curves of degree 4d+1 with 4 nests

M ‐curves of degree 9 with deep nests

“…Unfortunately, the word nest was used with different meanings by the several authors in the past. Take good care that the definition used here differs from the more standard one from [11], [16] or [4].…”

“…Our proof involves some classical tools: Bezout's theorem with auxiliary lines and conics, the Rokhlin-Mishachev formulas, Fiedler's theorem. We use supplementarily rational cubics, quartics (single curves or pencils of such curves), and Orevkov's complex orientation formulas from [16]. All of the arguments, and hence the statements, are also valuable for pseudo-holomorphic curves.…”

“…If N i , N j , N k have all depth d, call principal triangle p i p j p k the triangle p i p j p k that does not intersect J . The formulas hereafter are proven in [16] (with slightly different notations):…”