Polynomial differential equations with many real ovals in the same algebraic complex solution
Let FolR(2, d) be the space of real algebraic foliations of degree d in RP(2). For fixed d, let IntR(2, d) = {F 2 FolR(2, d) | F has a non-constant rational first integral}. Given F 2 IntR(2, d), with primitive first integral G, set O(F) = number of real ovals of the generic level (G = c). Let O(d) = sup{O(F) | F 2 IntR(2, d)}. The main purpose of this paper is to prove that O(d) = +1 for all d _ 5.
On compact analytic surfaces II
THEOREM 11.1. The family 9(,{, G) consists of all analytic fibre spaces Bi, '2C H'(A, f2(B#)). The notion of fibre spaces and their equivalences depends on the sheaf of structure groups.' By a (63-fibre space we mean a fibre space with a sheaf (3 of structure groups, and we say that two fibre spaces are (X-equivalent if they are equivalent as (63-fibre spaces. The fibre space Bi may be considered as an analytic fibre space, as an f2(BO)-fibre space, or as an f2(Bl)-fibre space. The cohomology class C) C H'(Ay, f(BO)) represents the f2(Bl)-equivalence class of Be. Let C& denote the fibre of BI over u c A. It is clear that
X. On the sextactic points of a plane curve
It is, in my memoir “On the Conic of Five-pointic Contact at any point of a Plane Curve”, remarked that as in a plane curve there are certain singular points, viz. the points of inflexion, where three consecutive points lie in a line, so there are singular points where six consecutive points of the curve lie in a conic; and such a singular point is there termed a “sextactic point.” The memoir in question (here cited as “former memoir”) contains the theory of the sextactic points of a cubic curve; but it is only recently that I have succeeded in establishing the theory for a curve of the order m. The result arrived at is that the number of sextactic points is = m(12m—27), the points in question being the intersections of the curve m with a curve of the order 12m—27, the equation of which is (12m2—54m+57)H Jac. (U, H, ΩH¯) +(m—2) (12m—27)H Jac. (U, H, ΩU¯) +40(m—2)2 Jac. (U, H, Ψ)=0, where U=0 is the equation of the given curve of the order m, H is the Hessian or determinant formed with the second differential coefficients (a, b, c, f, g, h) of U, and, (A, B, C, F, G, H) being the inverse coefficients (A=bc—f2, &c.), then Ω = (A, B, C, F, G, H)(∂x, ∂y, ∂z)2H, ψ = (A, B, C, F, G, H)(∂xH, ∂yH, ∂zH)2; and Jac. denotes the Jacobian or functional determinant, viz.