Chebyshev's method applied to polynomials with rotational symmetry

What the math gives to ML

The paper provides a sharp dynamical criterion for global behavior of Chebyshev's third-order root iteration on the rotationally symmetric family p_n(z)=z(z^n-1). Free critical points map radially by a constant factor D_n=(n-1)^2(2n+1)/(27(n+1)^2), producing a transition at n=17: critical values contract for n<=16 and expand for n>=17. This mechanism can be transferred to higher-order neural optimizers by treating stationarity along a training direction as a local polynomial root problem and using the maximum critical-orbit gain as a trust-region or step-size controller. The key experiment is to test whether the predicted critical-gain boundary separates stable third-order updates from exploding or basin-leaving updates.