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

Carlos Hermoso; Juan Gerardo Alcázar

paper

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

arXiv:2410.18867

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.