Characterization of polynomial surfaces of revolution and polynomial quadrics

TL;DR

This paper characterizes polynomial surfaces of revolution using associated plane curves, provides formulas for polynomial parametrizations over complex numbers, and explores real polynomial parametrizations, especially for quadrics.

Contribution

It offers a new characterization of polynomial surfaces of revolution and formulas for their polynomial parametrizations, advancing understanding of their real and complex cases.

Findings

Complete characterization of polynomial quadrics

Formulas for complex polynomial parametrizations

Initial analysis of real polynomial parametrizations

Abstract

In this paper, we characterize the polynomiality of surfaces of revolution by means of the polynomiality of an associated plane curve. In addition, if the surface of revolution is polynomial, we provide formulas for computing a polynomial parametrization, over $\mathbb{C}$, of the surface. Furthermore, we perform the first steps towards the analysis of the existence, and actual computation, of real polynomial parametrizations of surfaces of revolution. As a consequence, we give a complete picture of the real polynomiality of quadrics and we formulate a conjecture for the general case.