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.

Principle of generating critical points of rational functions and Gauss-Fourier conjecture for second-kind zeros
Lande Ma, Zhaokun Ma
arXiv21 Oct 2026PreprintCC BY-NC-ND 4.0Full textAnalysis & PDEsarXiv
Numerical experiments on the Hardy conjecture for the Gauss circle problem
Satoshi Yamaguchi, Shoichi Fujima, Shigehiko Kuratsubo +2
arXiv4 Sep 2026PreprintarXiv licenceFull textAlgebra & number theory Numerical analysis & scientific computingarXiv
A Fourier approach to Gromov's filling area conjecture
Le Chen, Xiaolong Li, Yimin Zhong
arXiv8 Sep 2026PreprintarXiv licenceFull textGeometry & topologyarXiv
On the Fourier Entropy-Influence Conjecture for Boolean Plateaued Functions
Vladimir N. Potapov
arXiv18 Sep 2026PreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv
Multiresolution Sobolev Trained Fourier Neural Operator for Cross-Resolution Surrogate Modelling of Parametric Two-Phase Darcy Flows in Porous Media
Zhao Zhang, Chengjin Guo, Zhenglong Chen +3
arXiv4 Oct 2026PreprintCC BY 4.0Full textNumerical analysis & scientific computingarXiv

Cite

BibTeX RIS

BibTeX

@misc{ma2026comprehensive,
  title = {{Comprehensive Resolution of the Gauss-Fourier Conjecture for the First Kind of Zeros}},
  author = {Lande Ma and Zhaokun Ma},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.30620},
  eprint = {2610.30620},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.30620},
  note = {CC BY-NC-ND 4.0},
}

RIS

TY  - UNPB
TI  - Comprehensive Resolution of the Gauss-Fourier Conjecture for the First Kind of Zeros
AU  - Lande Ma
AU  - Zhaokun Ma
PY  - 2026
DA  - 2026/10/22
PB  - arXiv
DO  - 10.48550/arxiv.2610.30620
UR  - https://arxiv.org/abs/2610.30620
AB  - 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.
ER  -

Abstract

Related

Cite