1.3 Proof strategy

The proof of Theorem 1.0.4 proceeds through several key steps:

1. Discrete orthogonality. We establish that sums of complex exponentials and cosines over the \(N\)-th roots of unity satisfy discrete orthogonality relations analogous to Fourier orthogonality.

2. Power sum invariance. Using binomial expansion of \(\cos ^j(\theta + 2\pi k/N)\) and the orthogonality relations, we prove that the power sums

   \begin{equation} \sum _{k=0}^{N-1} \cos ^j\left(\theta + \frac{2\pi k}{N}\right) \end{equation}

   13

   are independent of \(\theta \) for all \(1 \leq j {\lt} N\).

3. Newton’s identities. We apply Newton’s identities to show that if two sets of roots have identical power sums (except for the 0-th power sum, which counts the number of roots), their elementary symmetric polynomials match. Consequently, the monic polynomials with these roots have identical coefficients except possibly the constant term.

4. Chebyshev characterization. We identify the roots of \(T_N(x)\) as \(\cos ((2k + 1)\pi /(2N))\) for \(k = 0, \ldots , N-1\), and verify that these roots have the same power sums as the rotated roots \(r_k(\theta )\). This establishes coefficient-wise equality for degrees \(k \geq 1\).

5. Special case \(\theta = 0\). For \(\theta = 0\), explicit calculation shows \(S_N(x; 0) = T_N(x)\), confirming the identification up to the constant term.