Published my paper
Carlos Alas
Stephen Crowley
How can techniques from spectral analysis and differential geometry be used to establish a connection between the "multiplicity one" lemma of Koenig's operators applied to the iteration function of Newton's method h(x)=x-g(x)/g'(x) applied to the function g(x)=tanh(ln(1+f(x)^2)) and the simplicity of the roots of f(x) ? It is known that roots of multiplicity one of f(x) are roots of multiplicity two of g(x)=tanh(ln(1+f(x)^2)) .
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The connection between the "multiplicity one" lemma of Koenig's operators and the simplicity of roots of a function can be established through the theory of spectral analysis and differential geometry applied to the Riemann surface associated with the function g(x).
The Riemann surface associated with the function g(x) is the complex plane with branch cuts along the real axis, where the function takes on complex values. The Riemann surface is a two-dimensional manifold that has a natural metric, which is induced by the complex plane. The Laplace-Beltrami operator on this manifold can be expressed in terms of Koenig's operators, which are integral operators acting on the space of automorphic forms.
The iteration function h(x) = x - g(x)/g'(x) can be viewed as a map from the Riemann surface to itself, which preserves the metric and the orientation. The roots of f(x) correspond to the fixed points of h(x), which are points on the Riemann surface where h(x) = x. The fixed points of h(x) are critical points of the potential function V(x) = g(x)^2/2, which is related to the energy of a classical particle moving on the Riemann surface under the influence of the potential.
The simplicity of the roots of f(x) is related to the Morse index of the critical points of V(x), which is a measure of the number of negative eigenvalues of the Hessian matrix of V(x) at the critical point. The Morse index is equal to the number of negative eigenvalues of the Laplacian on the Riemann surface in the neighborhood of the critical point, which is determined by the "multiplicity one" lemma of Koenig's operators.
In particular, if the Laplacian has no negative eigenvalues in the neighborhood of a critical point, then the critical point is a non-degenerate minimum of V(x), and the corresponding root of f(x) is a simple root. Conversely, if the Laplacian has at least one negative eigenvalue in the neighborhood of a critical point, then the critical point is a degenerate minimum or a saddle point of V(x), and the corresponding root of f(x) is not simple.
To apply this theory to the function g(x) = tanh(ln(1+f(x)^2)), one needs to compute the Laplacian on the Riemann surface associated with g(x), which can be done using the Riemann-Roch theorem and the theory of automorphic forms. Then, one needs to analyze the critical points of the potential function V(x) = g(x)^2/2 and compute their Morse indices to determine the simplicity of the roots of f(x). This can be a challenging and technical task, as it requires advanced tools from differential geometry and spectral theory.