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An optimal multiplicity estimate for interpolation determinants in three 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

In the kit of Mignotte and Voutier for linear forms in three logarithms, the order of vanishing at the origin of the interpolation determinant is bounded below by Waldschmidt's general multiplicity lemma; the authors indicate that this estimate should be improvable. We give a lower bound for this order which is optimal for generic points. The functions involved span a space whose initial forms are described by a uniqueness property for a linear differential equation with constant coefficients. As a consequence, the left-hand side of their criterion (2.8) can be taken equal to (2K+1)(L−1)3log⁡ρ\frac{(2K+1)(L-1)}{3}\log\rho, with main term 2KL3log⁡ρ\frac{2KL}{3}\log\rho instead of KL2log⁡ρ\frac{KL}{2}\log\rho. On the two examples of their paper, with their public code modified in one line and the conditions of the zero lemma checked, the bounds given by the criterion go from 17.94⋅10617.94\cdot10^6 to 4.79⋅1064.79\cdot10^6 and from 28.85⋅10628.85\cdot10^6 to 7.00⋅1067.00\cdot10^6. The same estimate is the multiplicity input of our kit for three pp-adic logarithms.

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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  5. M. Mignotte, P. Voutier, A kit for linear forms in three logarithms, Math. Comp. 93 (2024), 1903--1951; arXiv:2205.08899.arXiv
  6. D. Niedbala Giraudin, A kit for linear forms in three pp-adic logarithms, preprint, 2026.
  7. 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