The Siegel-Shidlovskii Method Revisited
Abstract
We revisit the classical Siegel-Shidlovskii method in transcendental number theory from a new perspective, in which functional relations play a central role. Within a single self-contained framework, we recover the foundational results in the theory of E-functions, including the Siegel-Shidlovskii theorem, Beukers' lifting theorem, and André's theorem on the singularities of the minimal differential operator annihilating an E-function. Notably, the latter is obtained without any recourse to the theory of Gfunctions. This unified approach also allows us to resolve a problem posed by Shidlovskii: we obtain a general algebraic independence measure extending the classical Siegel-Lang measure to an arbitrary algebraically independent family of E-values. We also provide a short elementary proof of a second theorem of André, which has become an important tool in the theory of Efunctions and concerns the primality of the specialization, at a regular point, of the ideal of functional relations of a vector of E-functions satisfying a differential system.
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BibTeX
@misc{adamczewski2026siegelshidlovskii,
title = {{The Siegel-Shidlovskii Method Revisited}},
author = {Boris Adamczewski and Colin Faverjon},
howpublished = {HAL},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.02907},
eprint = {2611.02907},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05847695v1},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - The Siegel-Shidlovskii Method Revisited
AU - Boris Adamczewski
AU - Colin Faverjon
PY - 2026
DA - 2026/11/02
PB - HAL
DO - 10.48550/arxiv.2611.02907
UR - https://hal.science/hal-05847695v1
AB - We revisit the classical Siegel-Shidlovskii method in transcendental number theory from a new perspective, in which functional relations play a central role. Within a single self-contained framework, we recover the foundational results in the theory of E-functions, including the Siegel-Shidlovskii theorem, Beukers' lifting theorem, and André's theorem on the singularities of the minimal differential operator annihilating an E-function. Notably, the latter is obtained without any recourse to the theory of Gfunctions. This unified approach also allows us to resolve a problem posed by Shidlovskii: we obtain a general algebraic independence measure extending the classical Siegel-Lang measure to an arbitrary algebraically independent family of E-values. We also provide a short elementary proof of a second theorem of André, which has become an important tool in the theory of Efunctions and concerns the primality of the specialization, at a regular point, of the ideal of functional relations of a vector of E-functions satisfying a differential system.
ER -