An improved computation of component centers in the degree-\(n\) bifurcation set (Q1847588)

The governing equation locating component centers in the degree-\(n\) bifurcation set is a polynomial with very high degree, and its root-finding lacks numerical accuracy. The equation is transformed to have its degree reduced by a factor \((n-1)\). Newton's method applied to the transformed equation improves the accuracy with properly chosen initial values. The numerical implementation is done with Maple V using a large number of effective digits. Many cases are studied for \(2 \leq n \leq 25\). Remarkably improved results are obtained.