From Vieta to Chebyshev: A Unified Trigonometric Census of Polynomial Roots

Abstract

This article gives a trigonometric method for counting how many roots of a polynomial are real and how many are complex, without solving the equation. For a depressed polynomial $P(t)=t^d+c_{d-2}{t}^{d-2}+\cdots +c_{1}t+c_0$ of degree~$d$ with $c_{d-2}<0$ (the case $c_{d-2}\ge 0$ is discussed briefly in Section~\ref{sec:cgeq0} but not resolved), the substitution $t=u\cos \theta$, $u=2\sqrt{-c_{d-2}/d}$, matches $P$ against the Chebyshev identity $T_d(\cos \theta )=\cos d\theta$ and reduces the classification problem to a single trigonometric equation $f(\theta )={\sum }_{k=1}^{d-3}{\alpha }_k{\cos }^k\theta +\cos d\theta +\delta =0$ on $[0,\pi ]$. A bijection between the zeros of~$f$ and the roots of~$P$ in $[-u,u]$ gives an exact interior root count. For roots outside $[-u,u]$, a universal lemma based on the positivity of~$P^{(d-2)}$ gives an exact parity and an upper bound of $d-2$ roots per half-line; combined with the interior count, this pins down the total number of real roots exactly whenever at most one root lies on each exterior half-line, a condition we prove for the cubic, verify computationally throughout our examples, but show by an explicit quartic counterexample to fail in general for $d\ge 4$ (Remark~\ref{rem:counterexample}). We also compare the method's cost to Sturm's theorem (Section~\ref{sec:complexity}): it is not asymptotically cheaper, but avoids the coefficient blow-up of the Euclidean algorithm and applies via one uniform template to every degree $d\ge 4$, with the classical Vieta cubic ($d=3$) as the degenerate base case. The derivation is conceptually simple, unifying and extending the quartic and quintic analyses developed individually in earlier work.

arXiv commentImprovised and corrected a counterexample from the previous version. Under review