Polynomial potential minimization on the unit circle
Josiah Park
TL;DR
The paper tackles the problem of minimizing polynomial potentials on the unit circle by leveraging Chebyshev expansions and trigonometric moment reformulations, linking the problem to Toeplitz matrix conditions and semialgebraic optimization. It shows that, for rational coefficients, the optimal values are algebraic and extends the framework to energy minimization on spheres, where the optimum is likewise algebraic. It analyzes specific potentials such as t^{2k}+α C_{2k}^2(t) and the p-frame potential |t|^p, proposing a precise limiting relationship between their minimizers and highlighting the role of small discrete designs guided by Gegenbauer coefficients. The work also investigates minimizers on the circle, revealing non-uniqueness in some cases and proposing a derivative-root based conjecture for the support of optimal measures. Overall, the results provide an algebraic-structural perspective on discrete circle and sphere designs with implications for p-frame potentials and spherical codes.
Abstract
In the following, we study the minimization of polynomial potentials $ f(t) $ on the unit circle, where the potentials take the form \[ f(t) = \sum_{i=1}^n b_i x^{2i}, \quad b_i \in \mathbb{R}. \] This form arises in the context of truncations of expansions of $ p $-frame potentials. One approach to minimize these potentials involves rewriting the integral as a sum of integrals obtained by expanding the potential $ f(t) = \sum_{i=1}^n c_i T_i(t) $ in terms of Chebyshev polynomials. By replacing the inner product $ \langle x, y \rangle $ with $ \cos(θ_{x, y}) $, we can reformulate the original problem as: \[ \min_{μ\in P(T)} \int_T \int_T f(\langle x, y \rangle) dμ(x) dμ(y) \] as an equivalent form: \[ \min_{ν\in P([-π, π])} \sum_{i=1}^n c_i \int_{-π}^π\int_{-π}^π\cos(n(x - y)) dν(x) dν(y) \].