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PreprintarXiv licencearXiv

Root of the generic cubic as a power series in the discriminant

  • Jason Bland
  • Skip Garibaldi
  • Joel Rosenberg
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Abstract

An observation of J-P.~Serre implies that the generic monic cubic polynomial, unique among generic monic polynomials of degree at least two, has a root that is a power series in the discriminant; Serre asked for a formula. We give formulas that work over any field with an absolute value and in every characteristic. Over a complete non-archimedean field we identify the root intrinsically: it is the isolated root, the one farthest from the others. We also answer the next case of Serre's question, computing explicitly the distinguished ramified quadratic factor of the generic monic quartic. The methods combine Hensel's lemma, identities for generalized binomial series, and elementary non-archimedean analysis.

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Cite

BibTeXRIS
BibTeX
@misc{bland2026root,
  title = {{Root of the generic cubic as a power series in the discriminant}},
  author = {Jason Bland and Skip Garibaldi and Joel Rosenberg},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2605.25992},
  eprint = {2605.25992},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.25992},
}
RIS
TY  - UNPB
TI  - Root of the generic cubic as a power series in the discriminant
AU  - Jason Bland
AU  - Skip Garibaldi
AU  - Joel Rosenberg
PY  - 2026
DA  - 2026/10/20
PB  - arXiv
DO  - 10.48550/arxiv.2605.25992
UR  - https://arxiv.org/abs/2605.25992
AB  - An observation of J-P.~Serre implies that the generic monic cubic polynomial, unique among generic monic polynomials of degree at least two, has a root that is a power series in the discriminant; Serre asked for a formula. We give formulas that work over any field with an absolute value and in every characteristic. Over a complete non-archimedean field we identify the root intrinsically: it is the isolated root, the one farthest from the others. We also answer the next case of Serre's question, computing explicitly the distinguished ramified quadratic factor of the generic monic quartic. The methods combine Hensel's lemma, identities for generalized binomial series, and elementary non-archimedean analysis.
ER  -