Year: 2026
Type: preprint
Abstract: We classify graphs of complex polynomials in one variable up to isometry, both for the inner (length) distance and for the outer (Euclidean) distance of $\mathbb{C}^2\cong\mathbb{R}^4$. For polynomials $p,q$ with $°p\ge 2$ we show that the graphs of $p$ and $q$ are inner isometric if and only if they are outer isometric, if and only if $p(z)=α\,q(βz+h)+k$ or $p(z)=α\,\bar q(βz+h)+k$ for some $α,β,h,k\in\mathbb{C}$ with $|α|=|β|=1$, where $\bar q$ denotes the polynomial with conjugate coefficients. Moreover, every inner isometry between such graphs is the restriction of an isometry of $\mathbb{C}^2$; in this sense graphs of complex polynomials are isometrically rigid. We contrast this with the real case, where the two distances behave very differently: the graphs of any two real polynomials are inner isometric, whereas, for degree at least $2$, outer isometries are generated by translations, reflections in horizontal and vertical lines and half-turns.
Source: arXiv (Cornell University)
Institutions: Universidade Estadual do Ceará, Universidade Federal do Ceará
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7221806834