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

For over two centuries, the esteemed mathematicians Gauss and Fourier conjectured that each real critical point of the rational function $p^{{}^{\mathrm{\prime }}}(s)\mathrm{/}p(s)$ is associated with a pair of non-real zeros of the real polynomial $p(s)$. The conjecture remains unresolved. The real zeros of $(p^{{}^{\mathrm{\prime }}}(s)\mathrm{/}p(s){)}^{{}^{\mathrm{\prime }}}$ that are real multiple zero of $p^{{}^{\mathrm{\prime }}}(s)$ are defined as the second kind of zeros of $(p^{{}^{\mathrm{\prime }}}(s)\mathrm{/}p(s){)}^{{}^{\mathrm{\prime }}}$. In addition to the second kind of zeros, the other real zeros of $(p^{{}^{\mathrm{\prime }}}(s)\mathrm{/}p(s){)}^{{}^{\mathrm{\prime }}}$ are referred to as the first kind of zeros of $(p^{{}^{\mathrm{\prime }}}(s)\mathrm{/}p(s){)}^{{}^{\mathrm{\prime }}}$. We demonstrate that the first kind of zeros of $(p^{{}^{\mathrm{\prime }}}(s)\mathrm{/}p(s){)}^{{}^{\mathrm{\prime }}}$ 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 $(p^{{}^{\mathrm{\prime }}}(s)\mathrm{/}p(s){)}^{{}^{\mathrm{\prime }}}$ 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.