Inner and outer isometries of graphs of polynomials

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 $p(z)=\alpha q(\beta z+h)+k$ or $p(z)=\alpha \bar{q}(\beta z+h)+k$ for some $\alpha ,\beta ,h,k\in \mathbb{C}$ with $|\alpha |=|\beta |=1$, 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.