Generalize the polar-conjugate intersection scheme to three variables

Develop a geometric polar-based intersection algorithm that extends Paul de Casteljau’s conjugate-polar line method for intersections of two projective conics in the plane to systems of three polynomial equations in three variables, specifically to solving f(x) = g(x) = h(x) = 0 for x = [x, y, z] in projective three-space, with convergence properties analogous to the two-variable case.

Background

In Section 8, the paper reviews de Casteljau’s analytic polar form approach to root finding and his distinct geometric method for n = 2, where intersections of two projective conics are computed via iterative use of polars and conjugate points. Starting from a point, polars with respect to the two conics yield conjugate intersections and lines whose re-iteration produces points with quadratic and then quartic convergence toward the common root.

While the analytic scheme can be extended to higher dimensions, the geometric polar-conjugate method is only described for the two-variable conic case. The explicit statement that de Casteljau did not find a generalization to three variables highlights a concrete unresolved extension of the geometric procedure to systems f(x) = g(x) = h(x) = 0 in three variables.