Comprehensive Resolution of the Gauss-Fourier Conjecture for the First Kind of Zeros

Abstract

For over two centuries, the esteemed mathematicians Gauss and Fourier conjectured that each real critical point of the rational function $p^{'}(s)/p(s)$ is associated with a pair of non-real zeros of the real polynomial $p(s)$. The conjecture remains unresolved. The real zeros of $(p^{'}(s)/p(s))^{'}$ that are real multiple zero of $p^{'}(s)$ are defined as the second kind of zeros of $(p^{'}(s)/p(s))^{'}$. In addition to the second kind of zeros, the other real zeros of $(p^{'}(s)/p(s))^{'}$ are referred to as the first kind of zeros of $(p^{'}(s)/p(s))^{'}$. We demonstrate that the first kind of zeros of $(p^{'}(s)/p(s))^{'}$ emerge from the intersection of root loci emitted from non-real zeros of $p(s)$, thereby providing a comprehensive geometric interpretation of their relationship. Moreover, each first kind of zero of $(p^{'}(s)/p(s))^{'}$ is connected via root loci to a pair of conjugate non-real zeros of $p(s)$. This establishes the precise geometric relationship conjectured by Gauss and Fourier, fully resolving their original problem for first-kind zeros and completing a foundational chapter in polynomial theory. We have proven Gauss-Fourier conjecture about the first kind of zeros.