PreprintCC BY 4.0arXiv
The existence of small defining polynomials for number fields
Abstract
We prove that every number field of degree n and absolute discriminant D has a defining integral polynomial of degree n whose coefficients are all On(√D) in size, thus resolving a 1998 conjecture of Ruppert. Our proof compares ideal lattices with their trace duals, via which we handle all number fields with the exception of CM fields whose maximal totally real subfields have small discriminant. We then treat the remaining fields by averaging completed Epstein zeta functions over all possible ideal classes and archimedean scalings, for which we make critical use of Hecke's formula and the subconvexity theorem of Michel and~Venkatesh. Our proof does not use the Quasi-Riemann Hypothesis.
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BibTeX
@misc{bhargava2026existence,
title = {{The existence of small defining polynomials for number fields}},
author = {Manjul Bhargava and Sameera Vemulapalli},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.20600},
eprint = {2610.20600},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.20600},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The existence of small defining polynomials for number fields
AU - Manjul Bhargava
AU - Sameera Vemulapalli
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.20600
UR - https://arxiv.org/abs/2610.20600
AB - We prove that every number field of degree n and absolute discriminant D has a defining integral polynomial of degree n whose coefficients are all O_n(√D) in size, thus resolving a 1998 conjecture of Ruppert. Our proof compares ideal lattices with their trace duals, via which we handle all number fields with the exception of CM fields whose maximal totally real subfields have small discriminant. We then treat the remaining fields by averaging completed Epstein zeta functions over all possible ideal classes and archimedean scalings, for which we make critical use of Hecke's formula and the subconvexity theorem of Michel and~Venkatesh. Our proof does not use the Quasi-Riemann Hypothesis.
ER -