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A kit for linear forms in three pp-adic logarithms

Primarily AI-generated textHuman understanding: no partsmath.NT — Number Theory

Contributed by David Niedbala Giraudin ↗

Version 1 / Nov 02, 2026 / CC BY 4.0

Abstract

We transpose to the pp-adic case the kit of Mignotte and Voutier for linear forms in three logarithms. The new tool is a lower bound for the pp-adic valuation of interpolation determinants with no factorial loss: it is obtained by expanding the powers θℓz\theta^{\ell z} as binomial series in θ−1\theta-1, and then transporting an order of vanishing that we compute optimally for generic points, by means of the initial forms of the space of exponential polynomials involved. It holds for every prime pp. Combined with a rearrangement bound for the spread of the exponents and with Gouillon's zero lemma applied within a Teichmüller class, it gives a criterion of the form of criterion (2.8) of Mignotte–Voutier, in which log⁡ρ\log\rho is replaced by log⁡p\log p and the height term contains no analytic contribution. For two logarithms, in Qp\mathbb{Q}_p with pp odd, it improves Lemma 8 of Bugeaud–Laurent. A change of basis in the multiplicative group reduces to the case of principal units: the factor g=p−1g=p-1 leaves the volumes and moves into the heights. On examples, with exponents up to 101010^{10} or 103010^{30}, the complete bounds (degenerate cases included) are 154154 to 667667 times smaller than those of Yu's explicit theorem for p∈{5,13}p\in\{5,13\}, 608608 to 19801980 times for p∈{7,11}p\in\{7,11\} (where the explicit form of Yu goes through Q(i)\mathbb{Q}(i)), and 7373 to 105105 times for p=101p=101.

Provenance statement

The research and the writing were carried out by the author with Claude (Anthropic). The proofs were checked by independent AI reviews and by exact computation. No professional mathematician has reviewed the note yet.

Tools used

Anthropic
Claude Opus

References

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Show all 11 referencesShow fewer references
  1. K. Yu, pp-adic logarithmic forms and group varieties III, Forum Math. 19 (2007), 187--280.

Version history

  1. v1Submitted by David Niedbala GiraudinInitial depositCurrentOct 31, 2026