On the geometry of real polynomials of the 7-th degree with simple complex critical points
Abstract
We consider real monic polynomials with simple complex critical points. Critical values of such polynomial p can be considered as vertices of the uniquely defined polygon, symmetric with respect to the x-axis. The inverse image of this polygon is a tree-like system of ovals that touch each other externally in critical points. This system of ovals will be called the geometrical code of p. Let a1+i· b1,…,an+i· bn, a1<a2<…<an and bi>0, i=1,…,n, be the set of critical points of our polynomial p in the upper halfplane, and c1+i· d1,…,cn+i· dn be their images. The order of numbers ck and the list of signs of numbers dk will be called the algebraic code of p. In our experimental work we study geometrical and algebraic codes of the 7-th degree polynomials and the relations between them.
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BibTeX
@misc{khokhlova2026geometry,
title = {{On the geometry of real polynomials of the 7-th degree with simple complex critical points}},
author = {Elizaveta Khokhlova and Yury Kochetkov},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.06695},
eprint = {2611.06695},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.06695},
}
RIS
TY - UNPB
TI - On the geometry of real polynomials of the 7-th degree with simple complex critical points
AU - Elizaveta Khokhlova
AU - Yury Kochetkov
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.06695
UR - https://arxiv.org/abs/2611.06695
AB - We consider real monic polynomials with simple complex critical points. Critical values of such polynomial p can be considered as vertices of the uniquely defined polygon, symmetric with respect to the x-axis. The inverse image of this polygon is a tree-like system of ovals that touch each other externally in critical points. This system of ovals will be called the geometrical code of p. Let a_1+i· b_1,…,a_n+i· b_n, a_1<a_2<…<a_n and b_i>0, i=1,…,n, be the set of critical points of our polynomial p in the upper halfplane, and c_1+i· d_1,…,c_n+i· d_n be their images. The order of numbers c_k and the list of signs of numbers d_k will be called the algebraic code of p. In our experimental work we study geometrical and algebraic codes of the 7-th degree polynomials and the relations between them.
ER -