Peer-reviewedCC BY 4.0L'Enseignement Mathématique
On deciding transcendence of power series
Abstract
It is well known that algebraic power series are differentially finite (D-finite): they satisfy linear differential equations with polynomial coefficients. The converse problem---determining whether a given D-finite power series is algebraic or transcendental---is notoriously difficult. We prove that this problem is decidable: we give two theoretical algorithms and a transcendence test that results in an efficient Maple implementation.
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BibTeX
@article{bostan2026deciding,
title = {{On deciding transcendence of power series}},
author = {Alin Bostan and Bruno Salvy and Michael F. Singer},
journal = {L'Enseignement Mathématique},
year = {2026},
month = oct,
url = {https://hal.science/hal-05172426v2},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - On deciding transcendence of power series
AU - Alin Bostan
AU - Bruno Salvy
AU - Michael F. Singer
PY - 2026
DA - 2026/10/27
JO - L'Enseignement Mathématique
UR - https://hal.science/hal-05172426v2
AB - It is well known that algebraic power series are differentially finite (D-finite): they satisfy linear differential equations with polynomial coefficients. The converse problem---determining whether a given D-finite power series is algebraic or transcendental---is notoriously difficult. We prove that this problem is decidable: we give two theoretical algorithms and a transcendence test that results in an efficient Maple implementation.
SN - 0013-8584
SN - 2309-4672
ER -