Principle of generating critical points of rational functions and Gauss-Fourier conjecture for second-kind zeros

The precise origin of critical points in rational functions remains a longstanding open problem. This paper elucidates the mechanism underlying critical point generation in rational functions. For over two centuries, a conjecture attributed to Gauss and Fourier has remained unresolved: every real zero of $(p^{{}^{\mathrm{\prime }}}(s)\mathrm{/}p(s){)}^{{}^{\mathrm{\prime }}}$ is associated with a pair of non-real zeros of the real polynomial $p(s)$. The real zeros of $(p^{{}^{\mathrm{\prime }}}(s)\mathrm{/}p(s){)}^{{}^{\mathrm{\prime }}}$ that are multiple real zeros of $p^{{}^{\mathrm{\prime }}}(s)$ are termed second-kind zeros of $(p^{{}^{\mathrm{\prime }}}(s)\mathrm{/}p(s){)}^{{}^{\mathrm{\prime }}}$. By developing the root locus method, we establish a necessary and sufficient condition for the emergence of second-kind zeros: a second-kind zero $M$ arises if and only if at least two root loci of $2q\pi$ or $2q\pi +\pi$ degree, emitted from at least one pair of non-real conjugate zeros of $p(s)$ (in the extended complex plane, excluding the real axis), directly intersect another root locus at $M$ on the real axis. Our results resolve one conjecture of the Gauss-Fourier conjectures and reveal the fundamental mechanism by which critical points of rational functions are generated.