Nonzero Constant Wronskians of Polynomials and Laurent Polynomials, and Geometric Consequences

TL;DR

This paper characterizes polynomials and Laurent polynomials with nonzero constant Wronskians and explores geometric implications, including impossibility results and open questions for rational functions.

Contribution

It provides a complete characterization of polynomials and Laurent polynomials with constant Wronskians and extends the analysis to rational functions, proposing new conjectures.

Findings

Characterization of polynomials with nonzero constant Wronskian

Extension to Laurent polynomials with the same property

Impossibility results and open questions for rational functions

Abstract

We characterize the polynomials $p_1(t), ... , p_n(t)$ whose Wronskian $W(p_1, ... , p_n)$ is a nonzero constant. Then, we generalize our results to characterize the Laurent polynomials with the same property. Finally, for rational functions we prove an impossibility result for $n=2$, and pose the case $n \geq 3$ as an open question, although we suggest an impossibility conjecture. Some geometric consequences are derived, especially in the case of polynomials.