Principle of generating critical points of rational functions and Gauss-Fourier conjecture for second-kind zeros
Abstract
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'(s)/p(s))' is associated with a pair of non-real zeros of the real polynomial p(s). The real zeros of (p'(s)/p(s))' that are multiple real zeros of p'(s) are termed second-kind zeros of (p'(s)/p(s))'. 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 2qπ or 2qπ+π 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.
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BibTeX
@misc{ma2026principle,
title = {{Principle of generating critical points of rational functions and Gauss-Fourier conjecture for second-kind zeros}},
author = {Lande Ma and Zhaokun Ma},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.29790},
eprint = {2610.29790},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.29790},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Principle of generating critical points of rational functions and Gauss-Fourier conjecture for second-kind zeros
AU - Lande Ma
AU - Zhaokun Ma
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.29790
UR - https://arxiv.org/abs/2610.29790
AB - 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^'(s)/p(s))^' is associated with a pair of non-real zeros of the real polynomial p(s). The real zeros of (p^'(s)/p(s))^' that are multiple real zeros of p^'(s) are termed second-kind zeros of (p^'(s)/p(s))^'. 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 2qπ or 2qπ+π 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.
ER -