Computing the Newtonian Graph

In his study of Newton's root approximation method, Smale (1985) deened the Newto-nian graph of a complex univariate polynomial f. The vertices of this graph are the roots of f and f 0 and the edges are the degenerate curves of ow of the Newtonian vector eld N f (z) = ?f(z)=f 0 (z). The embedded edges of this graph form the boundaries of root basins in Newton's root approximation method. The graph deenes a treelike relation on the roots of f and f 0 , similar to the linear order when f has only real roots. We give an eecient algebraic algorithm based on cell decomposition to compute the Newtonian graph. The resulting structure can be used to query whether two points in C are in the same basin. This gives us a modiied version of Newton's method in which one can test whether a step has crossed a basin boundary. Steff ansson (1995) has recently extended this algorithm to handle rational and algebraic functions without signiicant increase in complexity. He has shown that the New-tonian graph tesselates the associated Riemann surface and can be used in conjunction with Euler's formula to give an NC algorithm to calculate the genus of an algebraic curve.