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\title{A kit for linear forms in three $p$-adic logarithms}
\author{David Niedbala Giraudin\\\small ORCID 0009-0009-1526-1178}
\date{Version 4, 30 October 2026}

\begin{document}
\maketitle

\begin{abstract}
We transpose to the $p$-adic case the kit of Mignotte and Voutier for linear forms in three
logarithms. The new tool is a lower bound for the $p$-adic valuation of interpolation determinants
with no factorial loss: it is obtained by expanding the powers $\theta^{\ell z}$ as binomial series
in $\theta-1$, 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 $p$. Combined with a rearrangement bound for the spread of the exponents and with
Gouillon's zero lemma applied within a Teichm\"uller class, it gives a criterion of the form of
criterion (2.8) of Mignotte--Voutier, in which $\log\rho$ is replaced by $\log p$ and the height term
contains no analytic contribution. For two logarithms, in $\mathbb Q_p$ with $p$ 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 $g=p-1$ leaves the volumes and moves into the heights. On examples, with
exponents up to $10^{10}$ or $10^{30}$, the complete bounds (degenerate cases included) are $154$ to
$667$ times smaller than those of Yu's explicit theorem for $p\in\{5,13\}$, $608$ to $1980$ times for
$p\in\{7,11\}$ (where the explicit form of Yu goes through $\mathbb Q(i)$), and $73$ to $105$ times
for $p=101$.
\end{abstract}

\section{Results}

Explicit lower bounds for linear forms in logarithms come in two forms: general theorems, in
$\log B$ (Matveev in the complex case, Yu in the $p$-adic case), and methods of interpolation
determinants, in $(\log B)^2$ but with much smaller constants, which serve to reduce a first bound.
For two logarithms, the latter exist in both cases (Laurent \cite{L08}; Bugeaud--Laurent \cite{BL},
then Chim \cite{C} with a dependence in $\log B$). For three logarithms, Mignotte and Voutier
\cite{MV} gave a kit for the complex case. To our knowledge there was none for the $p$-adic case;
we give one here.

\subsection*{Setting}
Let $p$ be a prime, $\alpha_1,\alpha_2,\alpha_3\ge2$ integers prime to $p$, multiplicatively
independent, and $b_1,b_2,b_3\ge1$ integers with $\gcd(b_1,b_2,b_3)=1$ and $p\nmid b_3$. Put
\[
\Lambda=\alpha_1^{b_1}\alpha_2^{b_2}-\alpha_3^{b_3}\ne0,\qquad
\beta_1=\frac{b_1}{b_3},\quad\beta_2=\frac{b_2}{b_3},\qquad g=\begin{cases}p-1 & p\ne2,\\ 1 & p=2.\end{cases}
\]
Write $d_1=\gcd(b_1,b_3)$, $d_2=\gcd(b_2,b_3)$, $b_3'=b_3/d_1$, $b_1'=b_1/d_1$,
$b_3''=b_3/d_2$, $b_2''=b_2/d_2$. For $p$ odd, write $\alpha_i=\zeta^{n_i}\theta_i$, where $\zeta$ is a
primitive $(p-1)$-th root of unity in $\Zp$, $n_i\in\mathbb Z$ and $\theta_i\in1+p\Zp$; for $p=2$,
take $n_i=0$ and $\theta_i=\alpha_i\in1+2\mathbb Z_2$.

Let $K,L\ge1$ be integers, $N=K(K+1)L/2$, and consider the family of the $N$ indices
$(k_i,m_i,\ell_i)$ with $k_i+m_i\le K-1$ and $0\le\ell_i\le L-1$; each value of $\ell$ is taken
$K(K+1)/2$ times. For $N$ distinct triples $(r_j,s_j,t_j)$ of non-negative integers, put
\[
\Delta=\det\Bigl(\tbinom{r_jb_3'+t_jb_1'}{k_i}\tbinom{s_jb_3''+t_jb_2''}{m_i}\,
\alpha_1^{\ell_ir_j}\alpha_2^{\ell_is_j}\alpha_3^{\ell_it_j}\Bigr)_{1\le i,j\le N}\in\mathbb Z,
\]
which is the determinant (2.6) of \cite{MV}. Let $D_V$ be the multiset
$\{a+c:0\le a\le K-1,\ 0\le c\le(K-a)L-1\}$, of cardinality $N$, and, for $0\le I\le N$, let
$\Theta_V(I)$ be the sum of the $I$ smallest elements of $D_V$. This number is at least the quantity
$\Theta(K-1,I)$ of \cite[(3.1)]{MV} used in Waldschmidt's multiplicity lemma.

\begin{theorem}[analytic upper bound with no loss]\label{thm:an}
Assume $n_1r_j+n_2s_j+n_3t_j\equiv c\pmod g$ for one $c$ and every $j$. Then
\[
v_p(\Delta)\ \ge\ \min_{0\le I\le N}\Bigl((N-I)\,v_p(\Lambda)+\Theta_V(I)\Bigr)-\sum_{i=1}^N(k_i+m_i).
\]
If moreover $v_p(\Lambda)\ge KL-1$, then
\[
v_p(\Delta)\ \ge\ \frac N2\cdot\frac{(2K+1)(L-1)}3 .
\]
\end{theorem}

With Waldschmidt's bound, the main term would be $\frac{N^2}{2K}$, that of Proposition~3.7 of
\cite{MV}; here it is asymptotically $\frac{N^2}{2K}\cdot\frac43$. No loss of the type $v_p(n!)$
occurs, and no hypothesis on the size of $K$ or $L$ relative to $p$ is required.

Put
\[
E(K,L)=\frac{(2K+1)(L-1)}3,\qquad
\lambda_L=\frac{2\lfloor L^2/4\rfloor}{L},
\]
and, as in (2.5) of \cite{MV},
\[
b=\frac{(R-1)b_3+(T-1)b_1}{2d_1}\cdot\frac{(S-1)b_3+(T-1)b_2}{2d_2}\cdot
\Bigl(\prod_{k=1}^{K-1}(k!)^{K-k}\Bigr)^{-12/(K(K-1)(K+1))}.
\]

\begin{theorem}[criterion for three $p$-adic logarithms]\label{thm:kit}
Let $K\ge2$, $L\ge2$, $R_j,S_j,T_j\ge1$ ($j=1,2,3$) be integers, $R>R_1+R_2+R_3$,
$S>S_1+S_2+S_3$, $T>T_1+T_2+T_3$, $\chi>0$, $V=\sqrt{(R_1+1)(S_1+1)(T_1+1)}$ and
$\mathcal M=\max\{R_1+S_1+1,S_1+T_1+1,R_1+T_1+1,\chi V\}$. Assume
\begin{align*}
(\mathrm{Z}1)\quad &(R_1+1)(S_1+1)(T_1+1)>g\,K\,\mathcal M\quad\text{and}\quad (R_1+1)(S_1+1)(T_1+1)>g\,L,\\
(\mathrm{Z}2)\quad &(R_2+1)(S_2+1)(T_2+1)>g\max\{2KL,\,K^2\},\\
(\mathrm{Z}3)\quad &(R_3+1)(S_3+1)(T_3+1)>3g\,K^2L,
\end{align*}
\begin{equation}\label{eq:crit}
E(K,L)\log p\ >\ \log N+\frac{2(K-1)}3\log b+\lambda_L\bigl((R-1)\log\alpha_1+(S-1)\log\alpha_2+(T-1)\log\alpha_3\bigr).
\end{equation}
Then $v_p(\Lambda)\le KL-2$, unless one of the conditions (2.14), (2.15) of \cite{MV} holds.
\end{theorem}

Recall that (2.14) is $b_1\le\max\{R_1,R_2\}$, $b_2\le\max\{S_1,S_2\}$, $b_3\le\max\{T_1,T_2\}$;
in that case $v_p(\Lambda)\le\log|\Lambda|/\log p\le(b_1\log\alpha_1+b_2\log\alpha_2+b_3\log\alpha_3)/\log p$.
Condition (2.15) is the existence of coprime integers $(u_1,u_2,u_3)\ne0$ with
$u_1b_1+u_2b_2+u_3b_3=0$ and $|u_1|\le U_1$, $|u_2|\le U_2$, $|u_3|\le U_3$, where
$U_1=\frac{(S_1+1)(T_1+1)}{\mathcal M-\max\{S_1,T_1\}}$, etc. \cite[(2.15)]{MV}; it reduces to
two logarithms (Section~\ref{sec:deg}).

Compared with (2.8) of \cite{MV}, criterion \eqref{eq:crit} replaces $\log\rho$ by $\log p$, and the
term $gL(a_1R+a_2S+a_3T)$, where $a_i\ge\rho|\log\alpha_i|+\log\alpha_i$ in the real case, by
$\lambda_L\sum(R-1)\log\alpha_i\approx\frac L4\sum R\cdot 2\log\alpha_i$: there is no analytic term
in $\rho|\log\alpha_i|$; moreover, the main term of the left-hand side is $\frac{2KL}3$ instead of
$\frac{KL}2$. The price is the factor $g$ in the volumes of the boxes. The hypothesis $p\nmid b_3$
concerns the exponent of the isolated term; Yu's theorem does not require it.

\begin{corollary}[two logarithms]\label{cor:2}
Let $p$ be a prime, $\gamma_1=a_1/c_1$, $\gamma_2=a_2/c_2$ two multiplicatively independent
rationals with $p\nmid a_1c_1a_2c_2$, $h_i\ge\log\max\{|a_i|,|c_i|\}$, and $b_1,b_2\ge1$ integers with
$p\nmid b_2$. Put $\Lambda_2=\gamma_1^{b_1}-\gamma_2^{b_2}$ and $\gamma_i=\zeta^{n_i}\theta_i$ as
above ($n_i=0$ if $p=2$). Let $\Lambda^*\subset\mathbb Z^2$ be a sublattice of index $g$ contained in
$\{(r,s):n_1r+n_2s\equiv0\pmod g\}$ (one always exists). Let $K\ge3$, $L\ge2$,
$R_1,R_2,S_1,S_2\ge1$, $R=R_1+R_2-1$, $S=S_1+S_2-1$, $N=KL$ and
$b=\frac{(R-1)b_2+(S-1)b_1}2\bigl(\prod_{k=1}^{K-1}k!\bigr)^{-2/(K^2-K)}$. Assume that there are
translates $C_1,C_2$ of $\Lambda^*$ such that
\begin{gather*}
\#\{\gamma_1^r\gamma_2^s:r<R_1,\,s<S_1,\,(r,s)\in C_1\}\ge L,\\
\#\{b_2r+b_1s:r<R_2,\,s<S_2,\,(r,s)\in C_2\}>(K-1)L,
\end{gather*}
and that
\[
K(L-1)\log p>\log N+(K-1)\log b+\kappa_1h_1+\kappa_2h_2,
\]
where $\kappa_1=\min\bigl\{\lambda_L(R-1),\ \frac{L(R+g-1)}2-\frac{gNL}{6(S+g-1)}\bigr\}$ and
$\kappa_2=\min\bigl\{\lambda_L(S-1),\ \frac{L(S+g-1)}2-\frac{gNL}{6(R+g-1)}\bigr\}$.
Then $v_p(\Lambda_2)\le KL-2$. In particular, the bound
$v_p(x^{p-1}-1)\le 69(p-1)\frac{\log y\log xy}{(\log p)^2}+4$ of \cite{N69}, which lowers Yamada's
constant $283$ \cite{Y}, holds with $3$ instead of $4$.
\end{corollary}

\section{Analytic bound (Theorem \ref{thm:an})}

\subsection{Setting up}
Assume $v_p(\Lambda)\ge1$, otherwise there is nothing to prove. Then
$\zeta^{n_3b_3}=\zeta^{n_1b_1+n_2b_2}$ and $\theta_3^{b_3}=\theta_1^{b_1}\theta_2^{b_2}(1+\varepsilon)$
with $1+\varepsilon=1-\Lambda/(\alpha_1^{b_1}\alpha_2^{b_2})$, so $v_p(\varepsilon)=v_p(\Lambda)$.
Since $p\nmid b_3$, raising to the power $b_3$ is a bijection of $1+p\Zp$ (of $1+2\mathbb Z_2$ if
$p=2$), and $\theta_3=\theta_1^{\beta_1}\theta_2^{\beta_2}(1+\varepsilon)^{1/b_3}$, powers with exponent in
$\mathbb Z_{(p)}$ being defined by the binomial series. Put $z_{1,j}=r_j+t_j\beta_1$,
$z_{2,j}=s_j+t_j\beta_2\in\mathbb Z_{(p)}$, $X_j=b_3'z_{1,j}$, $Y_j=b_3''z_{2,j}\in\mathbb Z$. All
columns being in the class $c$, row $i$ contains the common factor $\zeta^{\ell_ic}$, of valuation
zero, and
\[
v_p(\Delta)=v_p\Bigl(\det\bigl(\tbinom{X_j}{k_i}\tbinom{Y_j}{m_i}\theta_1^{\ell_iz_{1,j}}\theta_2^{\ell_iz_{2,j}}
(1+\varepsilon)^{\ell_it_j/b_3}\bigr)\Bigr).
\]
Write $(1+\varepsilon)^{\ell_it_j/b_3}=1+\varepsilon\,\delta_{ij}$; since
$(1+\varepsilon)^{e}-1=\sum_{n\ge1}\binom{e}{n}\varepsilon^n$ for $e\in\Zp$, we have $\delta_{ij}\in\Zp$.
By multilinearity in the rows, the determinant equals
$\sum_{I\subset\{1,\dots,N\}}\varepsilon^{N-|I|}F_I(\theta_1-1,\theta_2-1)$, where $F_I(u_1,u_2)$ is the
determinant whose row $i$ is
$\binom{X_j}{k_i}\binom{Y_j}{m_i}(1+u_1)^{\ell_iz_{1,j}}(1+u_2)^{\ell_iz_{2,j}}$, multiplied entry by
entry by $\delta_{ij}$ if $i\notin I$.

\subsection{Integrality}
For $z\in\Zp$, $(1+u)^z=\sum_{a\ge0}\binom za u^a$ has its coefficients in $\Zp$. Hence
$F_I\in\Zp[[u_1,u_2]]$. If $F_I$ has order at least $A$ at the origin, the series converges at
$u=\theta-1$, since $v_p(\theta_i-1)\ge1$, and $v_p(F_I(\theta-1))\ge A$: each monomial of total
degree $d\ge A$ has valuation at least $d$.

\subsection{Order of vanishing}
Let $\vartheta\in\mathbb C^2\setminus\{0\}$, $\lambda_0,\dots,\lambda_{L-1}$ pairwise distinct, and
$V=\operatorname{Span}\{z_1^kz_2^me^{\lambda_\ell\vartheta\cdot z}:k+m\le K-1,\ \ell<L\}$.

\begin{proposition}[multiplicity \cite{NM}]\label{prop:mult}
Let $I\subset\{1,\dots,N\}$, $\xi_1,\dots,\xi_N\in\mathbb C^2$, and $\Psi(x)$ an $N\times N$
determinant whose rows of index $i\in I$ are $(\varphi_i(x\xi_j))_j$ with $\varphi_i\in V$, the
other rows having entries analytic in $x$. Then $\operatorname{ord}_0\Psi\ge\Theta_V(|I|)$.
\end{proposition}

\begin{proof}
Let $w=\vartheta\cdot z$ and $u$ a complementary linear form. Then
$V=\bigoplus_{a<K}u^aW_a$ with $W_a=\operatorname{Span}\{w^be^{\lambda_\ell w}:b\le K-1-a\}$, the
solution space of $\prod_\ell(\frac d{dw}-\lambda_\ell)^{K-a}y=0$, of order $n_a=(K-a)L$: a non-zero
element of $W_a$ has order at most $n_a-1$ at $w=0$ (uniqueness for the Cauchy problem). The initial
form of a non-zero $f=\sum_au^af_a\in V$ is therefore a combination of monomials $u^aw^c$ with
$c\le n_a-1$, with no cancellation between distinct indices $a$. For a subspace $W\subset V$ of
dimension $I'$, a basis of $W$ with independent initial forms has initial degrees forming a
sub-multiset of $D_V$; for arbitrary points, the corresponding determinant has order at least the
sum of these degrees, hence at least $\Theta_V(I')$. We conclude by a Laplace expansion of $\Psi$
along the rows with index in $I$ (the complementary minors are analytic).
\end{proof}

\begin{lemma}\label{lem:ord}
$F_I$ has order at least $\Theta_V(|I|)-\sum_{i=1}^N(k_i+m_i)$ at the origin.
\end{lemma}

\begin{proof}
Put $G_I(\omega_1,\omega_2)=F_I(e^{\omega_1}-1,e^{\omega_2}-1)\in\mathbb Q_p[[\omega]]$. Since
$(1+u)^z=\exp(z\log(1+u))$ in $\mathbb Q_p[[u]]$, row $i$ of $G_I$ is
$\binom{X_j}{k_i}\binom{Y_j}{m_i}\exp\bigl(\ell_i(\omega_1z_{1,j}+\omega_2z_{2,j})\bigr)$, multiplied by
$\delta_{ij}$ if $i\notin I$. The polynomial $\binom{b_3'z_1}{k}\binom{b_3''z_2}{m}$ is a combination of
monomials $z_1^{\mu_1}z_2^{\mu_2}$ with $\mu_1+\mu_2\le k+m$. By multilinearity, $G_I$ is a
combination of determinants $G_{I,\mu}$ whose row $i$ is
$z_{1,j}^{\mu_{i,1}}z_{2,j}^{\mu_{i,2}}\exp(\ell_i\,\omega\cdot z_j)$ (times $\delta_{ij}$ if $i\notin I$),
with $d_i:=\mu_{i,1}+\mu_{i,2}\le k_i+m_i\le K-1$. For such a determinant and
$\vartheta=(\vartheta_1,\vartheta_2)\in\mathbb C^2\setminus\{0\}$, the function
\[
\Psi(x)=\det\Bigl((x z_{1,j})^{\mu_{i,1}}(xz_{2,j})^{\mu_{i,2}}\exp\bigl(\ell_i(\vartheta_1xz_{1,j}+\vartheta_2xz_{2,j})\bigr)[\delta_{ij}]\Bigr)
=x^{\sum d_i}\,G_{I,\mu}(x\vartheta_1,x\vartheta_2)
\]
vanishes at $x=0$ to order at least $\Theta_V(|I|)$, by Proposition~\ref{prop:mult}
($\lambda_\ell=\ell$; the rows of index $i\in I$ lie in $V$, the others are multiplied by
constants). The $z_{i,j}$ are rational and the $\delta_{ij}$ enter polynomially: the identity, true
for arbitrary complex $\delta_{ij}$, holds formally. The homogeneous components of $G_{I,\mu}$ of
degree $<\Theta_V(|I|)-\sum d_i$ therefore vanish at every $\vartheta\ne0$, hence identically. Thus
$G_I$ has order at least $\Theta_V(|I|)-\sum(k_i+m_i)$. Finally
$F_I(u)=G_I(\log(1+u_1),\log(1+u_2))$, and substituting series with no constant term and with linear
part the identity does not decrease the order.
\end{proof}

\noindent\emph{End of the proof.} We obtain
$v_p(\Delta)\ge\min_I\bigl((N-|I|)v_p(\Lambda)+\Theta_V(|I|)\bigr)-\sum(k_i+m_i)$, for all
$K,L\ge1$. If $v_p(\Lambda)\ge KL-1$, since every element of $D_V$ is at most $KL-1$, the minimum is
attained at $I=N$, and $\Theta_V(N)-\sum_i(k_i+m_i)=\frac{N(2K+1)(L-1)}6$ by a direct computation
($\sum_i(k_i+m_i)=LK(K+1)(K-1)/3$). \qed

\begin{remark}
The first inequality of Theorem~\ref{thm:an} is attained in many small configurations: the
verifier finds equality there. For generic points and $I=N$, Proposition~\ref{prop:mult} is
optimal \cite{NM}.
\end{remark}

\section{Arithmetic bound}

\begin{lemma}[rearrangement]\label{lem:R}
Let $\ell_1,\dots,\ell_N$ take each value $0,\dots,L-1$ exactly $Q$ times ($N=QL$) and let
$r_1,\dots,r_N\in[0,R-1]$. For every permutation $\sigma$,
$\bigl|\sum_i\ell_ir_{\sigma(i)}-\frac{L-1}2\sum_jr_j\bigr|\le\frac{(R-1)Q\lfloor L^2/4\rfloor}2$.
\end{lemma}

\begin{proof}
Since $\sum_i(\ell_i-\frac{L-1}2)=0$, the difference equals
$\sum_i(\ell_i-\frac{L-1}2)(r_{\sigma(i)}-\frac{R-1}2)$, whose absolute value is at most
$\frac{R-1}2\,Q\sum_{\ell=0}^{L-1}|\ell-\frac{L-1}2|=\frac{R-1}2Q\lfloor L^2/4\rfloor$.
\end{proof}

\begin{proposition}\label{prop:liou}
If $\Delta\ne0$, then
\[
\frac2N\,v_p(\Delta)\log p\le\log N+\frac{2(K-1)}3\log b+\lambda_L\bigl((R-1)\log\alpha_1+(S-1)\log\alpha_2+(T-1)\log\alpha_3\bigr),
\]
where $R,S,T$ are strict upper bounds for the coordinates of the points.
\end{proposition}

\begin{proof}
As in Proposition 3.9 of \cite{MV}, $\Delta=\mathcal P(\alpha_1,\alpha_2,\alpha_3)$ with
$\mathcal P\in\mathbb Z[X_1,X_2,X_3]$, whose monomials have exponents
$(\sum\ell_ir_{\sigma(i)},\sum\ell_is_{\sigma(i)},\sum\ell_it_{\sigma(i)})$, and
$|\mathcal P|\le N^{N/2}b^{(K-1)N/3}$ for the sup norm on the torus. Lemma~\ref{lem:R}, with
$Q=K(K+1)/2$, places these exponents in intervals of lengths $2G_1,2G_2,2G_3$, where
$G_1=\frac{(R-1)K(K+1)\lfloor L^2/4\rfloor}4$, etc. Write $\Delta=\alpha^{U}\widetilde{\mathcal P}(\alpha)$,
where $U_i$ is the least exponent of $X_i$ among the monomials with non-zero coefficient;
$\widetilde{\mathcal P}\in\mathbb Z[X]$ has degrees at most $2G_i$ and $|\widetilde{\mathcal P}|=|\mathcal P|$.
The integer $\widetilde{\mathcal P}(\alpha)$ is non-zero, with the same valuation as $\Delta$. By the
maximum principle on the polydisc applied to $X^{2G}\widetilde{\mathcal P}(1/X)$, with $\alpha_i\ge1$,
$|\widetilde{\mathcal P}(\alpha)|\le|\mathcal P|\prod\alpha_i^{2G_i}$. We conclude with
$p^{v_p(\Delta)}\le|\widetilde{\mathcal P}(\alpha)|$ and $4G_1/N=\lambda_L(R-1)$.
\end{proof}

\section{Zero lemma within a class}

\begin{proposition}\label{prop:zero}
Under (Z1)--(Z3), and if neither (2.14) nor (2.15) of \cite{MV} holds, there is a class $c$ modulo
$g$ such that the matrix formed by the columns $(r,s,t)$, $0\le r<R$, $0\le s<S$, $0\le t<T$,
$n_1r+n_2s+n_3t\equiv c$, has rank $N$.
\end{proposition}

\begin{proof}
For $j=1,2,3$, let $B_j=[0,R_j]\times[0,S_j]\times[0,T_j]$ and let $c_j$ be a class containing the
largest number $C_j\ge|B_j|/g$ of points of $B_j$. Let $\Sigma_j$ be the image of the points of $B_j$
of class $c_j$ under $\psi(r,s,t)=(r+t\beta_1,s+t\beta_2,\alpha_1^r\alpha_2^s\alpha_3^t)$, and
$c=c_1+c_2+c_3$. Since $R>R_1+R_2+R_3$ (and likewise for $S,T$), $\Sigma_1+\Sigma_2+\Sigma_3$ is
contained in the image of the points of class $c$ of the large box. A linear relation between the
rows gives a non-zero polynomial $P(X_1,X_2,Y)$, of total degree at most $K-1$ in $(X_1,X_2)$ and of
degree at most $L-1$ in $Y$, vanishing on this image. We check the hypotheses (3.17)--(3.19) of
Proposition 3.11 of \cite{MV} (Gouillon's lemma \cite{G}; the matrix is integral, and we work over
$\mathbb C$) exactly as in Section~4 of \cite{MV}, replacing the cardinalities of the boxes by the
$C_j$.
\begin{itemize}
\item $\psi$ is injective: a collision would give $\alpha_3^{mb_3}=\alpha_1^{mb_1}\alpha_2^{mb_2}$
for some $m>0$, whence $\Lambda=0$.
\item The values $\alpha_1^r\alpha_2^s\alpha_3^t$ are pairwise distinct, by multiplicative
independence. Hence $\#\{y\}=C_j$, and (4.2), (4.3) of \cite{MV} follow from $C_1>L$ and $C_2>2KL$,
guaranteed by (Z1), (Z2).
\item Outside (2.14), $(r,s,t)\mapsto(r+t\beta_1,s+t\beta_2)$ is injective on $B_1$ and $B_2$
(Lemma 3.17 of \cite{MV}), whence (4.4) by $C_2>K^2$.
\item Outside (2.15), each fibre of $(x_1,x_2)\mapsto\lambda x_1+\mu x_2$ on the whole box $B_1$ has
fewer than $\mathcal M$ elements (Lemma 3.21 of \cite{MV}), a fortiori on its points of class $c_1$;
whence $\#\{\lambda x_1+\mu x_2\}\ge C_1/\mathcal M>K$ by (Z1).
\item $\#\Sigma_3=C_3>3K^2L$ by (Z3).
\end{itemize}
Hence $P=0$, a contradiction.
\end{proof}

\section{Proofs of Theorem \ref{thm:kit} and Corollary \ref{cor:2}}

\emph{Theorem \ref{thm:kit}.} By Proposition~\ref{prop:zero}, one can choose $N$ columns of class
$c$ forming a minor $\Delta\ne0$. If $v_p(\Lambda)\ge KL-1$, Theorem~\ref{thm:an} gives
$\frac2Nv_p(\Delta)\ge E(K,L)$, and Proposition~\ref{prop:liou} contradicts \eqref{eq:crit}.
Hence $v_p(\Lambda)\le KL-2$.

\emph{Corollary \ref{cor:2}.} We follow the proof of Theorem~1 of \cite{BL} with the powers
$\gamma_i^r$ themselves (without raising to the power $p^t$), the matrix
$\bigl(\binom{rb_2+sb_1}{k}\gamma_1^{\ell r}\gamma_2^{\ell s}\bigr)$ (rows $(k,\ell)$, $k<K$, $\ell<L$;
columns in the translate $C_1+C_2$ of $\Lambda^*$, which is contained in one class), and the zero
lemma of Lemmas 6--7 of \cite{BL}, which gives rank $N$ (it only requires $E_1+E_2\subset C_1+C_2$).
For the analytic bound, we proceed as in Theorem~\ref{thm:an}, with one variable
$z_j=r_j+s_jb_1/b_2$ ($p\nmid b_2$), with $\theta_2=\theta_1^{b_1/b_2}(1+\varepsilon)^{1/b_2}$,
$v_p(\varepsilon)=v_p(\Lambda_2)$. We expand each binomial into monomials $z^{\mu_i}$, $\mu_i\le k_i$,
and pass as in Lemma~\ref{lem:ord} to $\Psi(x)=x^{\sum\mu_i}G_{I,\mu}(x\vartheta)$, whose row $i$ is a
series $\sum_ac_{i,a}x^az_j^a$ (times $\delta_{ij}$ if $i\notin I$). Expanding by multilinearity, two
rows of $I$ assigned the same exponent $a$ are proportional; the non-zero terms therefore have
distinct exponents on $I$, and the order is at least $|I|(|I|-1)/2$. We obtain
$v_p(\Delta)\ge\min_I\bigl((N-I)v_p(\Lambda_2)+I(I-1)/2\bigr)-LK(K-1)/2$, which equals $NK(L-1)/2$
as soon as $v_p(\Lambda_2)\ge N-1$. For $p$ odd, this replaces the bound $NK(L-1)/2-N\log N/\log p$
of Lemma~8 of \cite{BL}. For the arithmetic bound, we use the bound
$|\mathcal P|\le N^{N/2}b^{(K-1)N/2}$ of Lemma~6 of \cite{LMN} (used in Lemma~11 of \cite{BL}), and we
bound the spread of the exponents in two ways. Lemma~\ref{lem:R} with $Q=K$ gives
$4G_1/N=\lambda_L(R-1)$. On the other hand, let $g'$ be the index of the projection of $\Lambda^*$ on
the first coordinate and $g''$ that of $\Lambda^*\cap(\{0\}\times\mathbb Z)$; then $g'g''=g$. The
first coordinates of the points of $C_1+C_2$ lie in a progression of difference $g'$, and each is
taken at most $\lfloor(S-1)/g''\rfloor+1$ times. The proof of Lemma~10 of \cite{BL} then applies word
for word (it uses the hypothesis $\gcd(m_1,m_2,g)=1$ only to obtain this structure) and gives
$4G_1/N\le\frac{L(R+g-1)}2-\frac{gNL}{6(S+g-1)}$. We take the minimum, and likewise for $G_2$. If
$A=c_1^{V_1-U_1}c_2^{V_2-U_2}\widetilde{\mathcal P}(\gamma_1,\gamma_2)\in\mathbb Z$ ($U_i,V_i$ the least
and largest exponents, $V_i-U_i\le 2G_i$), the maximum principle applied to the homogenised
polynomial gives $|A|\le|\mathcal P|\,e^{2G_1h_1+2G_2h_2}$, and $v_p(\Delta)=v_p(A)$. If
$v_p(\Lambda_2)\ge N-1$, we obtain the negation of the criterion; hence $v_p(\Lambda_2)\le N-2$.

In the bound for $v_p(x^{p-1}-1)$ of \cite{N69}, the chain is the same with $\log N$ instead of
$3\log N$ (which only helps) and the conclusion $KL-2$ instead of $KL-1$. With $L=5$ and
$K=\lfloor 69L\Gamma/25\rfloor+1$, we obtain $KL-2\le69\,\Gamma+3$. \qed

\section{Degenerate cases}\label{sec:deg}

Assume (2.15), with $u_3\ne0$. Put $x=\alpha_3^{b_3}\alpha_1^{-b_1}\alpha_2^{-b_2}$, so that
$v_p(\Lambda)=v_p(x-1)\le v_p(x^{u_3}-1)$. Since $u_3b_3=-u_1b_1-u_2b_2$,
\[
x^{u_3}=\gamma_1^{-b_1}\gamma_2^{-b_2},\qquad \gamma_1=\alpha_1^{u_3}\alpha_3^{u_1},\quad
\gamma_2=\alpha_2^{u_3}\alpha_3^{u_2},
\]
with $h(\gamma_1)\le U_3\log\alpha_1+U_1\log\alpha_3$ and $h(\gamma_2)\le U_3\log\alpha_2+U_2\log\alpha_3$;
$\gamma_1,\gamma_2$ are multiplicatively independent. If $u_3=0$, then $u_1u_2\ne0$ and
$x^{u_1}=(\alpha_1^{u_2}\alpha_2^{-u_1})^{b_2}(\alpha_3^{u_1})^{b_3}$. In both cases we are reduced to
a form in two logarithms of rationals, to which we apply the better of the two results listed
below. Corollary~\ref{cor:2} requires that $p$ does not divide the second exponent; if $p$ divides
the second but not the first, we exchange the roles of $\gamma_1$ and $\gamma_2$ as well as those of
the $R_j$ and the $S_j$, which leaves the criterion unchanged. If $p$ divides both remaining
exponents, we write the form as $y^{p^k}-1$ with $v_p(y^{p^k}-1)\le v_p(y-1)+k$ ($p$ odd), which costs
at most $\log B/\log p$; Theorem~3 of \cite{BL} does not need this. The two results are:
\begin{itemize}
\item Theorem~3 of \cite{BL};
\item Corollary~\ref{cor:2}, with boxes $R_1S_1>gL$, $R_2S_2>g(K-1)L$ and translates of $\Lambda^*$
chosen by pigeonhole. If $b_2r+b_1s$ is not injective on the box $R_2\times S_2$, then
$b_1/d\le R_2-1$ and $b_2/d\le S_2-1$ with $d=\gcd(b_1,b_2)$, and $\Lambda_2$ is, up to a unit,
$\gamma^d-1$ with $h(\gamma)\le(R_2-1)h_1+(S_2-1)h_2$. For $p$ odd, if $g_0$ is the order of $\gamma$
modulo $p$, then $v_p(\gamma^d-1)=0$ if $g_0\nmid d$, and otherwise
$v_p(\gamma^d-1)=v_p(\gamma^{g_0}-1)+v_p(d)\le\bigl((p-1)h(\gamma)+\log2+\log B\bigr)/\log p$.
\end{itemize}




\section{Change of basis: reduction to $g=1$}\label{sec:base}

Restricting to Teichm\"uller classes puts the factor $g$ into the volumes of the boxes
(Z1)--(Z3), and (Z1) combines it with $\chi$. It can be avoided by a change of basis in the
multiplicative group, at the price of larger heights.

Write $\omega$ for the Teichm\"uller character: $\omega(\alpha_i)=\zeta^{n_i}$.

\begin{proposition}\label{prop:base}
Let $p$ be odd, and let $n_1,n_2,n_3$ be the Teichm\"uller indices of $\alpha_1,\alpha_2,\alpha_3$.
Let $\Lambda_0=\{x\in\mathbb Z^3:n_1x_1+n_2x_2+n_3x_3\equiv0\pmod g\}$, let $e_1,e_2,e_3$ be a basis
of $\Lambda_0$, and put $\gamma_j=\alpha_1^{e_{j1}}\alpha_2^{e_{j2}}\alpha_3^{e_{j3}}\in\mathbb Q^\times$.
\begin{enumerate}
\item The $\gamma_j$ are multiplicatively independent and $\gamma_j\equiv1\pmod p$.
\item If $v_p(\Lambda)\ge1$, the vector $\beta=(b_1,b_2,-b_3)$ lies in $\Lambda_0$; write
$\beta=\sum_jb'_je_j$ with $b'_j\in\mathbb Z$. Then $|b'_j|\le\kappa'_jB$, where $B=\max b_i$ and
$\kappa'_j=\sum_k|(E^{-1})_{kj}|$, $E$ being the matrix with rows $e_j$.
\item $v_p(\Lambda)=v_p\bigl(\gamma_1^{b'_1}\gamma_2^{b'_2}\gamma_3^{b'_3}-1\bigr)$. If
$\delta=\gcd(b'_1,b'_2,b'_3)$, then $v_p(\Lambda)=v_p\bigl(\gamma^{b'/\delta}-1\bigr)+v_p(\delta)$.
\end{enumerate}
\end{proposition}

\begin{proof}
(1) $E$ is invertible; $\omega(\gamma_j)=\zeta^{n\cdot e_j}=1$, so $\gamma_j\in1+p\mathbb Z_p$.
(2) $\Lambda=\alpha_3^{b_3}(\alpha^{\beta}-1)$, and $v_p(\Lambda)\ge1$ gives
$\omega(\alpha^{\beta})=\zeta^{n\cdot\beta}=1$. (3) $\alpha^{\beta}=\gamma^{b'}$, and for
$y\in1+p\mathbb Z_p$, $p$ odd, $v_p(y^\delta-1)=v_p(y-1)+v_p(\delta)$.
\end{proof}

\begin{theorem}[version with $g=1$]\label{thm:kit1}
Let $\gamma_1,\gamma_2,\gamma_3$ be multiplicatively independent rationals $u_j/w_j$ with
$\gamma_j\equiv1\pmod p$ ($p$ any prime), $h_j\ge\log\max\{|u_j|,|w_j|\}$, and
$\Lambda'=\gamma_1^{\beta_1}\gamma_2^{\beta_2}-\gamma_3^{\beta_3}\ne0$ with $\beta_j\ge1$,
$\gcd(\beta_1,\beta_2,\beta_3)=1$, $p\nmid\beta_3$. Then the conclusion of Theorem~\ref{thm:kit}
holds for $\Lambda'$ when one takes $n_i=0$ and $g=1$ in (Z1)--(Z3), and $h_j$ instead of
$\log\alpha_j$ in \eqref{eq:crit}; in case (2.14), $v_p(\Lambda')\le(\log2+\sum_j\beta_jh_j)/\log p$.
\end{theorem}

\begin{proof}
Theorem~\ref{thm:an} only uses $\theta_j=\gamma_j\in1+p\mathbb Z_p$; all columns are in a single
class. The zero lemma (Proposition~\ref{prop:zero}) only uses multiplicative independence and
combinatorial properties of the exponents. In Proposition~\ref{prop:liou}, $\Delta\in\mathbb Z_{(p)}$,
and the integer $A=\prod_jw_j^{V_j-U_j}\widetilde{\mathcal P}(\gamma)$ satisfies $v_p(A)\ge v_p(\Delta)$;
the maximum principle on the polydisc, applied to the homogenised polynomial, gives
$|A|\le|\mathcal P|\,e^{\sum_j2G_jh_j}$, as in Corollary~\ref{cor:2}.
\end{proof}

Corollary~\ref{cor:2} holds likewise with $g$ replaced by the index of $\Lambda^*$; for principal
units one takes $\Lambda^*=\mathbb Z^2$ (index $1$), its proof using only
$g'g''=[\mathbb Z^2:\Lambda^*]$.

We apply Theorem~\ref{thm:kit1} to the form $\gamma^{b'/\delta}-1$. Since $b'/\delta$ is primitive,
one of its coordinates, say $j_0$, is prime to $p$. For $j\ne j_0$, replace $\gamma_j$ by
$\gamma_j^{-1}$ if $b'_j<0$; replace $\gamma_{j_0}$ by $\gamma_{j_0}^{-1}$ if $b'_{j_0}>0$. Up to a unit,
the form becomes $\gamma_a^{|b'_a|/\delta}\gamma_b^{|b'_b|/\delta}-\gamma_{j_0}^{|b'_{j_0}|/\delta}$. If one
$b'_j$ vanishes, it has two logarithms and falls under Corollary~\ref{cor:2} (index $1$) or
\cite[Th\'eor\`eme 3]{BL}; if two vanish, $v_p(\gamma_j^m-1)\le(h_j+\log2+\log m)/\log p$. Since we do
not know which coordinate is isolated, we take the maximum of the three bounds, and add
$\log(\max_j\kappa'_jB)/\log p$ for $v_p(\delta)$. This method requires neither $p\nmid b_3$ nor
$\gcd(b_1,b_2,b_3)=1$.

We choose the basis minimising the product of the heights $h_j$ among combinations of short vectors
of $\Lambda_0$. The factor $g$ then moves into the heights, whereas the method of classes pays it in
the volumes, and squared in (Z1) ($V_1>(gK\chi)^2$ roughly). It cannot disappear: if the $\alpha_i$
are pairwise coprime and $\gamma_j=u_j/w_j$ is written in lowest terms, then $u_jw_j$ is the
product of the $\alpha_i^{|e_{ji}|}$, so $h(\gamma_j)\ge\frac12\sum_i|e_{ji}|\log\alpha_i$, and
Hadamard's inequality (in the form $|\det M|\le\prod_j\sum_i|M_{ji}|$) gives
\[
h(\gamma_1)h(\gamma_2)h(\gamma_3)\ \ge\ \frac{[\mathbb Z^3:\Lambda_0]}8\,\log\alpha_1\log\alpha_2\log\alpha_3 .
\]
This lower bound needs the coprimality: when the $\alpha_i$ share prime factors, cancellations in
$\gamma_j$ can make the heights smaller. In the examples below, where the $\alpha_i$ are distinct
primes, the product of the heights is $1.2$ to $1.4$ times
$g\log\alpha_1\log\alpha_2\log\alpha_3$.

\begin{table}[h]
\centering\footnotesize
\begin{tabular}{@{}rccrrrr@{}}
\toprule
$p$ & $B$ & heights $h_j$ & classes (Section~\ref{sec:deg}) & change of basis & Yu \cite{Yu,AKMRVW} & ratio\\
\midrule
5 & $10^{10}$ & 1.79\ 1.95\ 2.08 & $1.04\cdot10^{7}$ & $4.62\cdot10^{6}$ & $3.08\cdot10^{9}$ & 667\\
5 & $10^{30}$ & 1.79\ 1.95\ 2.08 & $9.26\cdot10^{7}$ & $3.98\cdot10^{7}$ & $9.24\cdot10^{9}$ & 232\\
7 & $10^{10}$ & 2.08\ 2.20\ 2.30 & $6.37\cdot10^{6}$ & $2.68\cdot10^{6}$ & $4.27\cdot10^{9}$ & 1592\\
7 & $10^{30}$ & 2.08\ 2.20\ 2.30 & $5.54\cdot10^{7}$ & $2.11\cdot10^{7}$ & $1.28\cdot10^{10}$ & 608\\
11 & $10^{10}$ & 2.48\ 2.64\ 3.30 & $9.10\cdot10^{6}$ & $2.10\cdot10^{6}$ & $4.16\cdot10^{9}$ & 1980\\
11 & $10^{30}$ & 2.48\ 2.64\ 3.30 & $6.72\cdot10^{7}$ & $1.60\cdot10^{7}$ & $1.25\cdot10^{10}$ & 780\\
13 & $10^{10}$ & 2.64\ 2.77\ 3.30 & $9.62\cdot10^{6}$ & $1.75\cdot10^{6}$ & $6.37\cdot10^{8}$ & 364\\
13 & $10^{30}$ & 2.64\ 2.77\ 3.30 & $7.02\cdot10^{7}$ & $1.24\cdot10^{7}$ & $1.91\cdot10^{9}$ & 154\\
101 & $10^{10}$ & 4.68\ 4.85\ 8.88 & $5.27\cdot10^{8}$ & $2.33\cdot10^{6}$ & $2.43\cdot10^{8}$ & 105\\
101 & $10^{30}$ & 4.68\ 4.85\ 8.88 & $1.25\cdot10^{9}$ & $9.98\cdot10^{6}$ & $7.29\cdot10^{8}$ & 73\\
\bottomrule
\end{tabular}
\caption{Complete upper bounds for $v_p(\Lambda)$ for $\Lambda=\alpha_1^{b_1}\alpha_2^{b_2}-\alpha_3^{b_3}$,
$b_i\le B$ ($\alpha=(2,3,5)$ if $p=7$, $(2,3,7)$ otherwise), degenerate cases included; the column
``classes'' assumes $p\nmid b_3$, the column ``change of basis'' does not. Column ``classes'':
restriction to Teichm\"uller classes (Section~\ref{sec:deg}). Column ``change of basis'':
Proposition~\ref{prop:base}, then Theorem~\ref{thm:kit1}, maximum over the three choices of the
isolated coordinate. The bases, the $B'=\max_j\kappa'_jB$ and the parameters are in the verifier; for
instance, for $p=101$, $e_1=(-2,-3,1)$, $e_2=(-7,3,0)$, $e_3=(-5,1,4)$, of index $100$,
$\gamma=(7/108,\,27/128,\,7203/32)$ and $B'=0.48\,B$. The ratio is that of Yu to the change of basis.
Yu's theorem is taken in the explicit form of \cite[Theorem 3.1, Corollary 3.3]{AKMRVW} ($n=3$;
field $\mathbb Q$ for $p\equiv1\pmod4$, $\mathbb Q(i)$ for $p\equiv3\pmod4$, which penalises Yu by a
factor of order $p/2$ for $p=7$ and $11$).}
\end{table}

\section{Scope}
Proved here: Theorem~\ref{thm:an}, Lemma~\ref{lem:R}, Propositions~\ref{prop:mult},
\ref{prop:liou}, \ref{prop:zero} and~\ref{prop:base}, Corollary~\ref{cor:2}, Theorem~\ref{thm:kit1}
and their assembly. The change of basis of Proposition~\ref{prop:base} is an elementary reduction,
probably known; we have not seen it used with $p$-adic interpolation determinants, where one usually
restricts the points to a class \cite{BL}. Used as they stand:
Gouillon's zero lemma \cite{G}, in the form of Proposition~3.11 of \cite{MV}; Lemmas 3.5, 3.17, 3.21
and the bound for $|\mathcal P|$ (Proposition~3.9) of \cite{MV}; Lemmas 6--7, 9--10 (in the form
adapted to translates of $\Lambda^*$) and Theorem~3 of \cite{BL}; Lemma~4 and the bound for
$|\mathcal P|$ of Lemma~6 of \cite{LMN} (two logarithms). The refinement $-N/(12RST)$ of the spread
(the $g$ of \cite{MV}) is not used.

\section*{Verification}
The ancillary file \texttt{VERIF\_KIT3P.py} (Python and numpy, no argument, about 15 seconds)
checks:
\begin{itemize}
\item Proposition~\ref{prop:mult}: exact order, computed in $\mathbb F_q[[x]]$, of determinants at
generic points and at points of the lattice of the kit; closed formulas;
\item Theorem~\ref{thm:an} for three logarithms, on integer determinants computed exactly, for
$p=2$, $3$, $5$, $7$, $11$ and $13$, with an exact relation and with $v_p(\varepsilon)=1$, $2$, $4$
or $7$;
\item the analytic bound for two logarithms, on integer determinants computed exactly, for $p=2$,
$3$, $5$, $7$ and $11$ with an exact relation, and for $p=2$, $3$, $5$ and $7$ with
$v_p(\varepsilon)=1$, $2$, $3$ or $5$;
\item Proposition~\ref{prop:zero}: rank $N$ of the matrices restricted to each class, for minimal
boxes satisfying (Z1)--(Z3);
\item the bound for the spread in Corollary~\ref{cor:2} drawn from Lemma~10 of \cite{BL}, on
translates of all sublattices of index $g\le12$ (Hermite normal form), with an adversarial choice
of the points;
\item for the change of basis, in exact arithmetic: the basis, its index, $\gamma_j\equiv1\pmod p$,
the heights, the $\kappa'_j$, then the criterion and the degenerate cases for the three isolated
coordinates, and the cases where one $b'_j$ vanishes;
\item each row of the method of classes: (Z1)--(Z3) and \eqref{eq:crit} at the parameters
indicated, the bounds $U_i$, both cases of (2.15) by the better of \cite[Theorem 3]{BL} and
Corollary~\ref{cor:2}, the case (2.14), and Yu's constant.
\end{itemize}
The verifier prints its messages in French.

\paragraph{Acknowledgements.} Technical assistance: Claude (Anthropic).

\raggedright
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\bibitem{NM} D.~Niedbala Giraudin, An optimal multiplicity estimate for interpolation determinants
in three logarithms, preprint, 2026.
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\end{document}