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\title{\texorpdfstring{An optimal multiplicity estimate for interpolation determinants\\ in three logarithms}{An optimal multiplicity estimate for interpolation determinants in three logarithms}}
\author{David Niedbala Giraudin\\\small ORCID 0009-0009-1526-1178}
\date{}

\begin{document}
\maketitle

\begin{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 $\frac{(2K+1)(L-1)}3\log\rho$, with main term $\frac{2KL}3\log\rho$ instead of
$\frac{KL}2\log\rho$. 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
$17.94\cdot10^6$ to $4.79\cdot10^6$ and from $28.85\cdot10^6$ to $7.00\cdot10^6$. The same estimate
is the multiplicity input of our kit for three $p$-adic logarithms.
\end{abstract}

\section{Result}

Let $K,L\ge1$ be integers, $N=K(K+1)L/2$, $\vartheta\in\mathbb C^2\setminus\{0\}$ and
$\lambda_0,\dots,\lambda_{L-1}$ pairwise distinct complex numbers. Put
\[
V=\operatorname{Span}\bigl\{z_1^{k}z_2^{m}\,e^{\lambda_\ell\,\vartheta\cdot z}\ :\ k+m\le K-1,\ 0\le\ell\le L-1\bigr\},
\]
a space of entire functions of $z=(z_1,z_2)$, and let $D_V$ be the multiset
\[
D_V=\bigl\{a+c\ :\ 0\le a\le K-1,\ 0\le c\le (K-a)L-1\bigr\},
\]
of cardinality $\sum_{a}(K-a)L=N$. For $0\le I\le N$, let $\Theta_V(I)$ be the sum of the $I$
smallest elements of $D_V$, and $\Theta(K_0,I)$ the quantity of \cite[(3.1)]{MV}.

\begin{proposition}[multiplicity]\label{prop:mult}
Let $W\subset V$ be a subspace of dimension $I$, with basis $f_1,\dots,f_I$, and let
$\xi_1,\dots,\xi_I\in\mathbb C^2$. Then $x\mapsto\det\bigl(f_n(x\xi_j)\bigr)_{1\le n,j\le I}$
vanishes at $x=0$ to order at least $\Theta_V(I)$. We have $\Theta_V(I)\ge\Theta(K-1,I)$, and
\begin{gather*}
\Theta_V(N)=\frac{K(K^2-1)L}{6}+\frac{K(K+1)(2K+1)L^2}{12}-\frac{K(K+1)L}4,\\
\Theta_V(N)-\sum_{k+m\le K-1}L\,(k+m)=\frac{N(2K+1)(L-1)}{6}.
\end{gather*}
For $W=V$ and generic points, the order is exactly $\Theta_V(N)$.
\end{proposition}

For comparison, $\Theta(K-1,N)\sim N^2/(2K)\sim K^3L^2/8$, whereas $\Theta_V(N)\sim K^3L^2/6$: the
gain is a factor $4/3$ on the main term.

\begin{proof}
Let $w=\vartheta\cdot z$, and let $u$ be a linear form such that $(w,u)$ are coordinates. The
polynomials of degree at most $K-1$ in $z$ are those of degree at most $K-1$ in $(w,u)$, so
\[
V=\bigoplus_{a=0}^{K-1}u^a\,W_a,\qquad
W_a=\operatorname{Span}\{w^be^{\lambda_\ell w}\ :\ b\le K-1-a,\ \ell<L\}.
\]
The space $W_a$, of functions of one variable $w$, is contained in 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 solution
therefore has order at most $n_a-1$ at $w=0$: otherwise its first $n_a$ Taylor coefficients would
vanish, and the linear recurrence given by the equation, with leading coefficient $1$, would kill
all the others. For a non-zero $f=\sum_au^af_a(w)\in V$, the initial form (the non-zero homogeneous
component of lowest degree in $(w,u)$) is a combination of monomials $u^aw^{c}$ with
$c=\operatorname{ord}_0f_a\le n_a-1$; two distinct indices $a$ give distinct monomials, so there is
no cancellation. Thus the space $\operatorname{in}(V)$ of initial forms is contained in
$\operatorname{Span}\{u^aw^c:(a,c)\text{ as in }D_V\}$, and for every degree $d$,
$\dim\operatorname{in}(W)_d\le\#\{(a,c):a+c=d\}$.

Choose a basis $g_1,\dots,g_I$ of $W$ whose initial forms are linearly independent (Gaussian
elimination by initial degree); then $\det(f_n(x\xi_j))=c\,\det(g_n(x\xi_j))$ with $c\ne0$, and row
$n$ is divisible by $x^{\deg\operatorname{in}(g_n)}$, whatever the points. The order is therefore at
least $\sum_n\deg\operatorname{in}(g_n)$. Since the initial forms of the $g_n$ are independent, the
multiset of their degrees is a sub-multiset of $D_V$; their sum is therefore at least that of the
$I$ smallest elements of $D_V$. For $W=V$: $\dim\operatorname{in}(V)=\dim V=N=\#D_V$, so each
$\operatorname{in}(V)_d$ is the span of all the monomials $u^aw^c$ allowed in degree $d$, and the
$g_n$ can be chosen with initial forms exactly these $N$ monomials. The coefficient of
$x^{\Theta_V(N)}$ is then $\det\bigl(\operatorname{in}(g_n)(\xi_j)\bigr)$, a non-zero polynomial in
the points (distinct monomials), whence equality for generic points. (Equivalently, $D_V$ is the
sequence of degrees of the standard monomials of the ideal $\bigcap_\ell(u,w-\lambda_\ell)^K$, that
is, the Hilbert function of $L$ collinear points of multiplicity $K$, in duality with the ``least''
space of de Boor and Ron \cite{BR}.) Finally, the number of elements of $D_V$ of degree $d$ is at
most $\min(d,K-1)+1$, which gives $\Theta_V\ge\Theta(K-1,\cdot)$. The closed formulas follow from
$\sum_{a}\sum_{c<(K-a)L}(a+c)$ and from $\sum(k+m)=LK(K+1)(K-1)/3$.
\end{proof}

\begin{lemma}\label{lem:I}
Let $I\subset\{1,\dots,N\}$ and let $\Psi(x)$ be an $N\times N$ determinant whose rows of index
$i\in I$ are $\bigl(\varphi_i(x\xi_j)\bigr)_j$ with $\varphi_i\in V$, and whose other rows have
entries analytic in $x$. Then $\operatorname{ord}_0\Psi\ge\Theta_V(|I|)$.
\end{lemma}

\begin{proof}
Expand $\Psi$ by Laplace along the rows with index in $I$: each $|I|\times|I|$ minor taken from
these rows has order at least $\Theta_V(|I|)$ by Proposition~\ref{prop:mult} (or vanishes if the
$\varphi_i$, $i\in I$, are dependent), and each complementary minor is analytic.
\end{proof}

\section{Application to the complex kit of Mignotte--Voutier}

In \cite{MV}, for $i\in I$, row $i$ of the determinant $\Psi_I$ consists of the
$\varphi_i(x\eta_j,x\zeta_j)$, with
\[
\varphi_i(\eta,\zeta)=c_i\,\eta^{k_i}\zeta^{m_i}e^{\lambda_i(\eta\log\alpha_1+\zeta\log\alpha_2)},\qquad
k_i+m_i\le K-1,\quad \lambda_i=\ell_i-\tfrac{L-1}2 ;
\]
this is an element of $V$ with $\vartheta=(\log\alpha_1,\log\alpha_2)$. For $i\notin I$, the entries
are multiplied by analytic functions. Lemma~\ref{lem:I} therefore gives $J_I\ge\Theta_V(|I|)$,
instead of $J_I\ge\Theta(K-1,|I|)$ (Lemma 3.6 of \cite{MV}).

\begin{theorem}\label{thm:MV}
Theorem 2.1 of \cite{MV} remains true when the left-hand side of (2.8),
$\bigl(\frac{KL}2+\frac L2-0.37K-2\bigr)\log\rho$, is replaced by
\[
\frac{(2K+1)(L-1)}3\log\rho,
\]
or by the smaller quantity $\bigl(\frac{2KL}3+\frac L3-(0.37+\frac13)K-2\bigr)\log\rho$.
\end{theorem}

\begin{proof}
In the proof of Proposition 3.7 of \cite{MV}, the coefficient of $\log\rho$ is
$\frac{2N(K-1)}3-\min_I\bigl(KL(N-|I|)+J_I\bigr)$. With $J_I\ge\Theta_V(|I|)$, and since each
element of $D_V$ is at most $KL-1<KL$, this minimum is at least $\Theta_V(N)$, whence the
coefficient $-\frac N2\cdot\frac{(2K+1)(L-1)}3$; in \cite{MV}, it is bounded above by
$-\frac{N^2}{2K}\bigl(1-\frac2{3L}-\frac2{3KL}-\frac1{3L^2}-\frac{16}{3K^2L}\bigr)$. In the proof of
Proposition 3.10 of \cite{MV}, after division by $N/2$, this last quantity appears exactly as the
coefficient of $\log\rho$ in the intermediate inequality,
$\frac{KL}2+\frac L2-(\frac13+\frac1{6L})K-(\frac23+\frac3K+\frac1{6L}+\frac8{3K^2})$, which (2.8)
then bounds below by $\frac{KL}2+\frac L2-0.37K-2$; the other terms ($\log N!$, $N\log2$, $0.001$)
do not involve $\log\rho$. We replace it by $\frac{(2K+1)(L-1)}3$. The second form comes from
$\frac{(2K+1)(L-1)}3-\bigl(\frac{KL}2+\frac L2-0.37K-2\bigr)=\frac{KL-2K-L}6+\frac{11K}{300}+\frac53\ge\frac{KL-2K-L}6$.
\end{proof}

For the large values of $K$ and $L$ used in practice, the ratio of the two left-hand sides is about
$1.33$.

\subsection*{The examples of \cite{MV}}
The code of \cite{MV} (\texttt{github.com/PV-314/lfl3-kit}) computes (2.8) in the function
\texttt{get\_eqn42} of the file \texttt{utils-step3.gp}. For the computations below, we replaced
\begin{center}
\texttt{eqn42LHS=(bigK*bigL/2+bigL/2-0.37*bigK-2)*log(rho);}\\
by\quad\texttt{eqn42LHS=(bigK*bigL*2/3+bigL/3-(0.37+1/3)*bigK-2)*log(rho);}
\end{center}
that is, the second, weaker form of Theorem~\ref{thm:MV}. Moreover, this code does not check the
conditions (2.9)--(2.13) of the zero lemma outside debugging mode; we make this check mandatory
(function \texttt{check\_zero\_estimate}, completed by the comparison of condition (2.9)), in both
versions. The iterations start, as in \cite{MV}, from Matveev's bound, with the same search grids
for both versions, and are continued until they stabilise. The proofs of \cite{MV} assume at the
outset $p>10\cdot10^6$ (example 1) and $p>20\cdot10^6$ (example 2); a bound below this threshold
therefore gives, with no further work, $p\le10\cdot10^6$ and $p\le20\cdot10^6$. Going lower would
require redoing step (1) of \cite{MV} with a smaller threshold, which we do not do: these examples,
already solved elsewhere, serve here as benchmarks, and what matters is the ratio of the bounds
(about $3.7$ and $4.1$).

\begin{table}[ht]
\centering\small
\begin{tabular}{@{}lccc@{}}
\toprule
& stated in \cite{MV} & code of \cite{MV}, (2.9)--(2.13) checked & Theorem~\ref{thm:MV}\\
\midrule
Example 1 ($y^p=F_n$, \cite{BMS1}) & $p<17.92\cdot10^6$ & $17.94\cdot10^6$ & $4.79\cdot10^6$\\
Example 2 ($x^2+7=y^p$) & $p<25\cdot10^6$ & $28.85\cdot10^6$ & $7.00\cdot10^6$\\
\bottomrule
\end{tabular}
\end{table}

\section{Application to the \texorpdfstring{$p$}{p}-adic kit}

In our $p$-adic kit \cite{NK}, the analytic bound with no loss rests on the same multiplicity
lemma, applied to the formal series $G_{I,\mu}$. Lemma~\ref{lem:I} applies word for word, and one
obtains the following statement, which is Theorem~1 of \cite{NK}.

\begin{theorem}\label{thm:p}
With the notation of \cite{NK}, and under the class hypothesis of its Theorem 1,
\[
v_p(\Delta)\ \ge\ \min_{0\le I\le N}\bigl((N-I)\,v_p(\Lambda)+\Theta_V(I)\bigr)-\sum_{i=1}^N(k_i+m_i),
\]
and, if $v_p(\Lambda)\ge KL-1$, $v_p(\Delta)\ge\frac N2\cdot\frac{(2K+1)(L-1)}3$. Consequently, the
criterion of \cite{NK} holds with $E(K,L)=\frac{(2K+1)(L-1)}3$ and the conclusion
$v_p(\Lambda)\le KL-2$.
\end{theorem}

The hypotheses $K\ge3$, $L\ge5$ are not used here. The bound of Theorem~\ref{thm:p} is often
attained: over $150$ series of two or three integer determinants computed exactly, at least one
determinant of the series attains it in $44$ series.

\begin{table}[ht]
\centering\footnotesize
\begin{tabular}{@{}rcrrrr@{}}
\toprule
$p$ & $B$ & $(L,\,K,\,\chi)$ & total, Theorem~\ref{thm:p} & Yu \cite{Yu,AKMRVW} & ratio\\
\midrule
5 & $10^{10}$ & (133, 114\,445, 8) & $1.52\cdot10^{7}$ & $3.08\cdot10^{9}$ & 202\\
5 & $10^{30}$ & (521, 259\,645, 12) & $1.47\cdot10^{8}$ & $9.24\cdot10^{9}$ & 63\\
7 & $10^{10}$ & (106, 91\,045, 8) & $9.76\cdot10^{6}$ & $4.27\cdot10^{9}$ & 438\\
7 & $10^{30}$ & (415, 196\,589, 12) & $8.85\cdot10^{7}$ & $1.28\cdot10^{10}$ & 145\\
11 & $10^{10}$ & (133, 102\,041, 8) & $1.42\cdot10^{7}$ & $4.16\cdot10^{9}$ & 293\\
11 & $10^{30}$ & (210, 553\,429, 20) & $1.19\cdot10^{8}$ & $1.25\cdot10^{10}$ & 105\\
13 & $10^{10}$ & (133, 106\,032, 8) & $1.52\cdot10^{7}$ & $6.37\cdot10^{8}$ & 42\\
13 & $10^{30}$ & (235, 522\,896, 20) & $1.23\cdot10^{8}$ & $1.91\cdot10^{9}$ & 16\\
101 & $10^{10}$ & (149, 3\,384\,594, 30) & $5.27\cdot10^{8}$ & $2.43\cdot10^{8}$ & 0.46\\
101 & $10^{30}$ & (295, 4\,144\,076, 30) & $1.25\cdot10^{9}$ & $7.29\cdot10^{8}$ & 0.59\\
\bottomrule
\end{tabular}
\caption{Complete upper bounds for $v_p(\Lambda)$ for $\Lambda=\alpha_1^{b_1}\alpha_2^{b_2}-\alpha_3^{b_3}$,
$b_i\le B$, $p\nmid b_3$ ($\alpha=(2,3,5)$ if $p=7$, $(2,3,7)$ otherwise), degenerate cases
included, with the multiplicity estimate of Theorem~\ref{thm:p} and with the two-logarithm step of
the degenerate case (2.15) based on the rearrangement bound alone. The final version of \cite{NK}
also uses Lemma~10 of Bugeaud--Laurent on translates of sublattices in that step, which divides
these totals further by $1.46$ to $1.77$ for $p\le13$.}
\end{table}

\section{Remarks}

\begin{remark}
Mignotte and Voutier write \cite[Section~1.5 (2)]{MV}: ``Calculations suggest that the estimate
there should be roughly $\Theta(2K,|I|)$ instead of $\Theta(K,|I|)$. The main term on the
left-hand side of (2.8) would then become $KL$, rather than $KL/2$.'' Since $\Theta(K_0,I)$
decreases in $K_0$, we read this sentence as announcing an order of about twice as much; this is
an interpretation. Proposition~\ref{prop:mult} shows that, for generic points, the order has main
term $K^3L^2/6$, that is $4/3$ of Waldschmidt's bound; for generic points, this determinant
therefore cannot give more than the main term $2KL/3$. For the points of the lattice of the kit,
Proposition~\ref{prop:mult} is a lower bound; the verifier finds equality for small values of $K$
and $L$. The mechanism (uniqueness for differential equations with constant coefficients, Hilbert
function of multiple points) is classical; we have not found it applied to the multiplicity lemma
of interpolation determinants in three logarithms.
\end{remark}

\begin{remark}
For two logarithms, the argument gives the order $N(N-1)/2$, already used by Laurent and by
Laurent--Mignotte--Nesterenko \cite{LMN}: the exponential structure brings nothing more in that
case.
\end{remark}

\section*{Verification}
The ancillary file \texttt{VERIF\_MULT3.py} (Python and numpy, no argument, about ten seconds)
checks:
\begin{itemize}
\item the order of the determinants $\det(z^{\mu}e^{\ell\,x\vartheta\cdot z})$ for $W=V$, computed
exactly in $\mathbb F_q[[x]]$ ($q$ a $61$-bit prime), for six pairs $(K,L)$, at generic points and
at points of the lattice of the kit: it equals $\Theta_V(N)$;
\item the closed formulas for $1\le K,L<40$, and $\Theta_V\ge\Theta(K-1,\cdot)$ at about fifty
values of $I$ for each $(K,L)$;
\item Theorem~\ref{thm:p} on $150$ series of exact integer determinants, for $p=2$, $3$, $5$, $7$,
$11$ and $13$;
\item each row of the $p$-adic table.
\end{itemize}
The figures for the examples of \cite{MV} are obtained with their PARI/GP code, modified as
indicated; they are not recomputed by the ancillary file.

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

\raggedright
\begin{thebibliography}{9}
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A robust implementation for solving the $S$-unit equation and several applications,
arXiv:1903.00977.
\bibitem{BMS1} Y.~Bugeaud, M.~Mignotte, S.~Siksek, Classical and modular approaches to
exponential Diophantine equations I. Fibonacci and Lucas perfect powers, \emph{Ann. of Math.}
\textbf{163} (2006), 969--1018.
\bibitem{BR} C.~de Boor, A.~Ron, On multivariate polynomial interpolation, \emph{Constr. Approx.}
\textbf{6} (1990), 287--302.
\bibitem{LMN} M.~Laurent, M.~Mignotte, Y.~Nesterenko, Formes lin\'eaires en deux logarithmes et
d\'eterminants d'interpolation, \emph{J.~Number Theory} \textbf{55} (1995), 285--321.
\bibitem{MV} M.~Mignotte, P.~Voutier, A kit for linear forms in three logarithms, \emph{Math.
Comp.} \textbf{93} (2024), 1903--1951; arXiv:2205.08899.
\bibitem{NK} D.~Niedbala Giraudin, A kit for linear forms in three $p$-adic logarithms, preprint,
2026.
\bibitem{Yu} K.~Yu, $p$-adic logarithmic forms and group varieties III, \emph{Forum Math.}
\textbf{19} (2007), 187--280.
\end{thebibliography}
\end{document}