A short history of multivariate Wronskians
Abstract
The Wronskian is the classical tool for testing the linear independence of functions of one variable: a family of analytic functions is linearly dependent if and only if the determinant built on their successive derivatives vanishes identically. For functions of several variables, there is no canonical choice of partial derivatives, and multivariate generalizations of the Wronskian have been proposed, forgotten and rediscovered many times since the 1870s, each time in the service of a different application: invariant theory, nomography, projective differential geometry, Diophantine approximation, differential algebra, the arithmetic of curves over finite fields, symmetry methods for differential equations, and computer algebra. We retrace this history, from Rosanes and Pasch to the present day, relying on the original sources. We point out an early erroneous criterion, due to Pasta, together with the counterexample that refutes it, given by Kellogg in 1912. We observe that most of the correct proofs, from Ostrowski's in 1919 to recent ones, rest on a single idea: restricting the functions to a sufficiently generic curve, and using the classical Wronskian along this curve. Finally, we describe how the question of which multivariate Wronskians are really needed was only settled in the twenty-first century.
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BibTeX
@misc{bostan2026short,
title = {{A short history of multivariate Wronskians}},
author = {Alin Bostan and Philippe Dumas},
howpublished = {HAL},
year = {2026},
month = oct,
url = {https://hal.science/hal-05839886v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A short history of multivariate Wronskians
AU - Alin Bostan
AU - Philippe Dumas
PY - 2026
DA - 2026/10/26
PB - HAL
UR - https://hal.science/hal-05839886v1
AB - The Wronskian is the classical tool for testing the linear independence of functions of one variable: a family of analytic functions is linearly dependent if and only if the determinant built on their successive derivatives vanishes identically. For functions of several variables, there is no canonical choice of partial derivatives, and multivariate generalizations of the Wronskian have been proposed, forgotten and rediscovered many times since the 1870s, each time in the service of a different application: invariant theory, nomography, projective differential geometry, Diophantine approximation, differential algebra, the arithmetic of curves over finite fields, symmetry methods for differential equations, and computer algebra. We retrace this history, from Rosanes and Pasch to the present day, relying on the original sources. We point out an early erroneous criterion, due to Pasta, together with the counterexample that refutes it, given by Kellogg in 1912. We observe that most of the correct proofs, from Ostrowski's in 1919 to recent ones, rest on a single idea: restricting the functions to a sufficiently generic curve, and using the classical Wronskian along this curve. Finally, we describe how the question of which multivariate Wronskians are really needed was only settled in the twenty-first century.
ER -