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  pdftitle={A unimodular continued fraction for cubic irrationals},
  pdfauthor={Will Howe, Grant Huffman, Jensen Kohlmeyer, and Liam Kruer},
  pdfsubject={Hermite's problem on periodic expansions of cubic irrationals}
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\title{A unimodular continued fraction for cubic irrationals}
\author[W. Howe]{Will Howe}
\email{\href{mailto:howew@purdue.edu}{howew@purdue.edu}}
\author[G. Huffman]{Grant Huffman}
\email{\href{mailto:huffma97@purdue.edu}{huffma97@purdue.edu}}
\author[J. Kohlmeyer]{Jensen Kohlmeyer}
\email{\href{mailto:jenwin@purdue.edu}{jenwin@purdue.edu}}
\author[L. Kruer]{Liam Kruer}
\email{\href{mailto:kruerl@purdue.edu}{kruerl@purdue.edu}}
\date{November 11, 2026}
\subjclass[2020]{11J70, 11A55, 11R16}
\keywords{Hermite's problem, cubic irrationals, multidimensional continued fractions, units}

\begin{document}

\begin{abstract}
We construct a single algorithm that assigns to every real number a
recoverable sequence of positive integers which is eventually periodic
exactly when the input is a cubic irrational. Its matrix digits are positive
unimodular transformations on the two-dimensional simplex. Their products
determine nested rational triangles whose diameters decrease by at least a
factor of two at each step. An effectively specified partition combines
Euclidean reduction with the dominant eigenvectors of cubic units. In both
cubic signatures the matrix expansion eventually repeats a single matrix,
whose Perron eigenvalue is a unit generating the input field.
\end{abstract}

\maketitle

\section{Introduction}

In 1848, Hermite asked Jacobi for a characterization of cubic irrationals
by eventual periodicity, analogous to Lagrange's theorem for ordinary
continued fractions~\cite{hermite,karpenkov}. Karpenkov's periodicity
theorem covers the totally real case and lists the case of a real cubic
with two nonreal conjugates as Problem~4~\cite[\S\,9]{karpenkov}.
Tagnon gives exact periodicity certificates for a collection of inputs
in this latter signature~\cite{tagnon}.

We use the general projective formulation of multidimensional continued
fractions~\cite[\S\,2.1]{rada}. A fixed, countable partition selects
an integer matrix from the current vector. Here the partition is specified
by finite searches, including an enumeration of integer matrices. By an
exact comparison on a real input we mean a test of the sign, including
equality, of a rational polynomial evaluated at that input.

\begin{theorem}[Hermite's problem]\label{thm:hermite}
There is a deterministic rule assigning to each real number \(\alpha\)
an infinite sequence of positive integers which determines \(\alpha\).
Every digit can be determined in finitely many steps, and all
input-dependent comparisons have degree at most two in \(\alpha\).
The sequence is
eventually periodic if and only if \([\mathbb Q(\alpha):\mathbb Q]=3\).
In the cubic case the matrix digits are eventually constant, and the Perron
eigenvalue of the repeating matrix is a unit generating
\(\mathbb Q(\alpha)\).
\end{theorem}

\section{Matrix digits and convergence}

Let \(\Delta=\{x\in\mathbb R_{\ge0}^3:\sum_i x_i=1\}\),
let \(S=\relint\Delta\), and write \(\pi(v)=v/\sum_i v_i\) for a nonzero
nonnegative vector \(v\). A matrix \(M\) acts as a digit through the map
\[
 F_M(x)=\pi(M^{-1}x)
\]
on the set of \(x\in S\) for which \(M^{-1}x\) has strictly positive
coordinates. We use matrices \(M\in\SL_3(\mathbb Z)\) with strictly
positive entries satisfying
\begin{equation}\label{eq:cross-ratios}
 M_{ij}M_{k\ell}\le 2M_{i\ell}M_{kj}
 \qquad (1\le i,k,j,\ell\le 3).
\end{equation}
For any nonnegative invertible matrix \(Q\), let \(K(Q)=\pi(Q\Delta)\).
This is the convex hull of the normalized columns of \(Q\), since
normalizing a nonnegative linear combination of the columns gives a
convex combination of their normalizations, and conversely.

\begin{lemma}\label{lem:contraction}
For nonnegative invertible \(Q\) and any matrix digit \(M\) satisfying
\eqref{eq:cross-ratios},
\[
 K(QM)\subseteq K(Q)
 \quad\text{and}\quad
 \diam K(QM)\le \tfrac12\diam K(Q).
\]
\end{lemma}

\begin{proof}
Let \(c_i=\pi(Qe_i)\) and let \(s_i\) denote the sum of the entries of
column \(i\) of \(Q\). Column \(j\) of \(QM\), after normalization, is
\(\sum_i p_i^{(j)}c_i\), where
\[
 p_i^{(j)}=\frac{s_iM_{ij}}{\sum_h s_hM_{hj}}.
\]
This proves the inclusion. For fixed columns \(j,\ell\), set
\(r_i=M_{ij}/M_{i\ell}\). Condition \eqref{eq:cross-ratios} gives
\(\max_i r_i\le 2\min_i r_i\), so
\[
 \frac{p_i^{(j)}}{p_i^{(\ell)}}=
 \frac{r_i}{\sum_h p_h^{(\ell)}r_h}\ge\tfrac12.
\]
Thus \(p^{(j)}\) is the average of \(p^{(\ell)}\) and another probability
vector. The normalized columns \(j,\ell\) of \(QM\) are at distance
at most half the diameter of \(K(Q)\). The diameter of a convex hull
is the largest distance between two of its vertices, which proves the
claim.
\end{proof}

Suppose \(M_0,M_1,\ldots\) are successive digits of \(x\), and set
\(Q_n=M_0\cdots M_{n-1}\) with \(Q_0=I\). If \(x^{(n)}\) is the state
after \(n\) digits, then \(x=\pi(Q_nx^{(n)})\). In an infinite expansion
the triangles \(K(Q_n)\) contain \(x\) and have diameter at most \(2^{-n}\).
They recover \(x\) as their unique common point, with explicit rational
enclosures from their vertices.

\section{Euclidean reduction in small cones}

\begin{lemma}\label{lem:small-cone}
Let \(x\in S\), and let \(D\subset S\) be a closed triangle with rational
vertices and \(x\in\relint D\). In finitely many steps one can find
a positive \(B\in\SL_3(\mathbb Z)\) with \(K(B)\subseteq D\) and
\(B^{-1}x>0\), or produce a nonzero rational row \(c\) with \(cx=0\).
The tests involving \(x\) are signs of rational linear forms.
\end{lemma}

\begin{proof}
The cone over \(D\) is the intersection of three homogeneous half-spaces
with integer defining rows. Start at \(B=I\) and impose these inequalities
one at a time on its columns. For the current inequality \(\ell v\ge0\),
write \(b_i=Be_i\), \(a_i=\ell b_i\), and \(t=B^{-1}x\). Maintain \(t>0\).

If the \(a_i\) have both signs, choose indices \(i,j\) with
\(a_i=P>0\) maximal and \(a_j=-N<0\) minimal. If \(t_i>t_j\), replace
\(b_j\) by \(b_j+b_i\). This replaces \(t_i\) by \(t_i-t_j\) and leaves
the other coordinates of \(t\) unchanged. If \(t_j>t_i\), replace \(b_i\)
by \(b_i+b_j\). If \(t_i=t_j\), the difference of the two corresponding
rows of \(B^{-1}\) gives the desired relation.

A replacement changes one extreme value to \(P-N\). As long as the
signs stay mixed, \(P\) and \(N\) never increase. If their sum is unchanged,
the number of entries equal to \(P\) or \(-N\) decreases. Hence the pair
consisting of their sum and this count decreases lexicographically.
The loop terminates. Since \(\sum_i a_i t_i=\ell x>0\), its successful
termination leaves all \(a_i\ge0\). Column additions preserve the
inequalities already imposed, as well as determinant one. After the
third inequality, every column is in the cone over \(D\). Since
\(D\subset S\), every column is strictly positive.
\end{proof}

\begin{lemma}\label{lem:routing}
Under the hypotheses of Lemma~\ref{lem:small-cone}, suppose that each
ratio \(v_i/v_k\) varies by a factor of at most two on \(D\). Given any
rational \(\eta>0\), there is a finite procedure returning either a
nonzero rational relation on \(x\) or a matrix digit \(M\) satisfying
\eqref{eq:cross-ratios} and
\begin{equation}\label{eq:routing}
 K(M)\subseteq D,\qquad
 M^{-1}x>0,\qquad
 \tau(F_M(x))<\eta,\qquad \tau(v)=v_1+v_2.
\end{equation}
\end{lemma}

\begin{proof}
Use Lemma~\ref{lem:small-cone} to obtain \(B\) or a relation. In the
former case apply Euclidean division to the first two coordinates of
\(B^{-1}x\), keeping the third fixed. A division replaces the larger
coordinate by its remainder modulo the smaller coordinate. The update
is effected by the inverse of a nonnegative unimodular elementary
matrix. A zero remainder gives a nonzero rational linear relation on
\(x\), since the coordinate change is invertible and rational.

If no remainder vanishes, the successive remainders tend to zero.
Indeed, their decreasing limit \(L\) satisfies \(L\ge 2L\) by the division
recurrence. Since the third coordinate stays positive, finitely many
divisions yield \eqref{eq:routing} for \(M=BE\), where \(E\) is the
product of the elementary matrices. Each column of \(M\) is a nonzero
nonnegative combination of columns of \(B\), so \(M\) is positive and
\(K(M)\subseteq K(B)\). The ratio bound on \(D\) applied to the
normalized columns of \(M\) gives \eqref{eq:cross-ratios}.
\end{proof}

In applying Lemma~\ref{lem:routing} one can choose \(D\) with any
prescribed positive upper bound on its diameter. One can also require
that it lie in a specified open half-space containing \(x\).
Sufficiently small triangles about \(x\) satisfy the ratio condition by
continuity, and their vertices can be chosen rational. Enumerating
rational triangles in height order finds one in finitely many steps.
The ratio and diameter bounds can be tested on the vertices. The
linear-fractional ratio \(v_i/v_k\) is bounded between its extreme vertex
values, since its value on a convex combination is a weighted average
of the vertex ratios with weights proportional to the denominator
coordinates. The half-space tests needed below have algebraic
coefficients fixed during the construction.

\section{Cubic directions and fixed points}

Call \(x\in S\) a \emph{cubic direction} if
\(1,x_2/x_1,x_3/x_1\) form a \(\mathbb Q\)-basis of a real cubic field.
This property is invariant under invertible rational projective
changes of coordinates whenever the new direction lies in \(S\).
The change of coordinates replaces the three entries of a field basis
by another basis, and normalization divides them by one nonzero
element of the same field.

\begin{lemma}\label{lem:units}
The cubic directions are exactly the positive eigenvector directions
of matrices \(A\in\SL_3(\mathbb Z)\) with irreducible characteristic
polynomial and an eigenvalue \(\lambda>1\) strictly dominant in modulus.
In this statement the positive eigenvector is for \(\lambda\).
Such pairs \((x,A)\), with \(x\in S\), admit an effective enumeration.
\end{lemma}

\begin{proof}
Given a cubic direction, put \(b=(1,x_2/x_1,x_3/x_1)\tr\),
\(K=\mathbb Q(b_1,b_2,b_3)\), and \(L=\sum_i\mathbb Z b_i\).
By Dirichlet's unit theorem, the logarithmic vectors of the units of
\(\mathcal O_K\) form a full lattice in the weighted sum-zero
hyperplane. The open cone in this hyperplane with a positive component
at the chosen real embedding and negative components at all other
places contains a lattice point. To see this, follow a ray in the
interior of the cone and approximate it to bounded distance by lattice
points. Squaring the resulting unit makes its real value positive
and its norm one.

Choose \(m\ge1\) with \(m\mathcal O_K L\subseteq L\) by clearing the
denominators of the products of an integral basis of \(\mathcal O_K\)
with the \(b_i\), expressed in the basis \(b\). A further positive
power \(u\) of the unit is \(1\) modulo \(m\mathcal O_K\). The same is
true of \(u^{-1}\), so multiplication by \(u\) preserves \(L\)
bijectively. Define \(A\) by \(u b_i=\sum_j A_{ij}b_j\). It is integral,
has determinant \(\operatorname{Norm}_{K/\mathbb Q}(u)=1\), and
satisfies \(Ab=ub\). Its eigenvalues are the three conjugates of \(u\).
The element \(u>1\) is a unit and cannot be rational, so it has degree
three and the characteristic polynomial is irreducible.

Conversely, for a matrix \(A\) as stated, choose an eigenvector for
\(\lambda\) over \(\mathbb Q(\lambda)\). Its images under the three
field embeddings are eigenvectors for distinct eigenvalues of \(A\)
and are linearly independent over \(\mathbb C\). Any rational linear
relation on the coordinates of the first vector would hold on all
three, which is impossible. Dividing by its first coordinate gives a
basis of \(\mathbb Q(\lambda)\), and hence the normalized positive
direction is cubic.

Enumerate the integer matrices in order of increasing maximum entry
in absolute value, using a fixed tie order. Determinant, polynomial
irreducibility, strict spectral dominance, and the sign pattern of
an eigenvector are finite tests in exact algebraic arithmetic.
Normalize the positive eigenvectors to obtain the list.
\end{proof}

\pagebreak
\begin{lemma}\label{lem:fixed}
Let \(x\in S\) be a cubic direction with matrix \(A\) from
Lemma~\ref{lem:units}. For any rational \(\eta>0\) one can find two
matrix digits \(M,P\), both satisfying \eqref{eq:cross-ratios}, and a
point \(y\in S\) with
\[
 y=F_M(x),\qquad \tau(y)<\eta,\qquad F_P(y)=y.
\]
\end{lemma}

\begin{proof}
Take a left eigenvector \(\ell\) of \(A\) for \(\lambda\) with
\(\ell x=1\). Such a normalization exists because the dominant
eigenvalue is simple. Apply Lemma~\ref{lem:routing} with \(D\)
contained in \(\ell v>0\). Rational independence rules out a relation,
and every entry of \(\ell M\) is positive. Set \(H=M^{-1}AM\). Its
normalized powers satisfy
\[
 \lambda^{-k}H^k\longrightarrow (M^{-1}x)(\ell M).
\]
The limit is positive and has rank one. Therefore \(H^k\) is positive
and satisfies \eqref{eq:cross-ratios} for all sufficiently large \(k\).
Choose the first such power \(P\) by exact integer tests. It is integral
and unimodular, and \(y=\pi(M^{-1}x)\) is its positive eigenvector
direction.
\end{proof}

\section{The partition of the simplex}

Fix the enumeration in Lemma~\ref{lem:units} and a deterministic
ordering for every subsequent finite search. Starting with
\(\eta_0=1\), define rational numbers \(0<\eta_{s+1}<\eta_s/2\) and
a rule on the bands
\[
 \mathcal B_s=\{x\in S:\eta_{s+1}\le\tau(x)<\eta_s\},
 \qquad s=0,1,2,\ldots.
\]
A band has an ordinary branch and may have two special branches.
The ordinary branch moves to \(\tau<\eta_{s+1}\) or returns a
rational relation. The special branches stay in the band.

At stage \(s\), take the \(s\)-th listed direction, counting from zero.
Use the already constructed bands to follow this direction until it
reaches \(\tau<\eta_s\) or a previously specified fixed point. The
finite residence property proved below ensures that the simulation
terminates. If it reaches a fixed point, leave the two special
branches empty and set \(\eta_{s+1}=\eta_s/3\).

Otherwise let \(x\) be the entry point into \(\tau<\eta_s\). Carry
its matrix \(A\) along the simulation by conjugation, replacing \(A\)
by \(N^{-1}AN\) at each step with digit \(N\). Apply
Lemma~\ref{lem:fixed} with a rational \(\eta<\tau(x)/2\) to get
\(M,P,y\). Choose open rational triangles \(U,V\) in \(S\) with
\(x\in U\), \(y\in V\), and the following properties.
\begin{enumerate}
 \item \(F_M\) is defined on \(\overline U\), and
       \(F_M(\overline U)\subset V\).
 \item \(F_P\) is defined on \(\overline V\), and
       \(F_P(\overline V)\cap\overline U=\varnothing\).
 \item \(\overline U,\overline V,F_P(\overline V)\) are compact subsets
       of \(\{v\in S:\tau(v)<\eta_s\}\), with
       \(\overline U\cap\overline V=\varnothing\).
\end{enumerate}
To make these choices, use \(x\ne y\) and \(F_P(y)=y\). First take \(V\)
small enough that both its closure and its image avoid a neighborhood
of \(x\). Then take \(U\) small enough that \(F_M(\overline U)\subset V\).
Both searches use rational triangles and finite algebraic tests. A
projective image of a closed triangle on the positive domain of either
map is the triangle spanned by the images of the vertices. Choose
\(\eta_{s+1}\) less than the minimum of \(\tau\) on the three compact
sets and less than \(\eta_s/2\).

The band rule emits \(M\) on \(U\) and \(P\) on \(V\), and applies the
corresponding map. On every other point of the band apply
Lemma~\ref{lem:routing} with target \(\eta_{s+1}\). In this ordinary
branch also require
\begin{equation}\label{eq:ordinary-size}
 \diam D<\frac{1}{[3(s+1)]^2}.
\end{equation}
Any matrix digit returned by the ordinary branch then has an entry
greater than \(s+1\). If all its entries were at most \(s+1\), every
normalized column would have denominator at most \(3(s+1)\). Two
distinct normalized columns would differ in some coordinate by at
least \(1/[3(s+1)]^2\). Invertibility makes the normalized columns
distinct, and their containment in \(D\) contradicts
\eqref{eq:ordinary-size}.

\pagebreak
\begin{proposition}\label{prop:global-rule}
Every stage of the partition is constructed in finitely many steps.
The resulting rule depends only on the current point of \(S\).
Every cubic direction has an infinite, eventually constant matrix
expansion. For a noncubic direction the rule either returns a rational
relation after finitely many digits or has an infinite expansion with
unbounded matrix entries.
\end{proposition}

\begin{proof}
In a completed band, the branch on \(U\) sends a point into \(V\).
The branch on \(V\) stays in the band and cannot send a point into
\(U\). A point that stays in \(V\) forever lies in every \(K(P^n)\).
By Lemma~\ref{lem:contraction}, their intersection is the single
fixed point \(y\). A point in \(V\) other than \(y\) thus leaves \(V\)
in finite time and takes the ordinary branch. Consequently an orbit
leaves a completed band in finite time unless it halts or reaches
the fixed point.

When constructing stage \(s\), the simulated direction is cubic
throughout its orbit, so rational independence rules out a halt.
The simulation visits the earlier bands in strictly increasing order
as it leaves them. It therefore reaches a known fixed point or
\(\tau<\eta_s\) in finite time. Exact algebraic equality suffices to
recognize the known fixed points in this simulation. This proves the
induction, including the first stage, for which there are no earlier
bands.

For an arbitrary current point, only finitely many bands must be
constructed to locate it because \(\tau(x)>0\) and \(\eta_s\to0\).
The data of those bands depend solely on the fixed enumeration.
Evaluating the rule on the point uses rational linear tests, including
the floor tests in Euclidean division. Each finite sequence of these
tests specifies a piece by homogeneous rational linear conditions.
This gives the countable partition underlying the matrix expansion.

Every cubic direction occurs in the enumeration. At its stage it
either reaches a previous fixed point or is sent by the new branch
from \(x\) to \(y\), where its digit remains \(P\). A noncubic direction
cannot reach a cubic fixed point, by invariance under the invertible
rational moves. An infinite orbit of such a direction must take
ordinary branches in bands of arbitrarily large index. The entry
bound from \eqref{eq:ordinary-size} proves the last claim.
\end{proof}

\section{Expansion of a real number}

\begin{proof}[Proof of Theorem~\ref{thm:hermite}]
For \(\alpha\ne0\), record its sign. Set \(b=|\alpha|\) and initialize
the rule at \(x=\pi((1,b,b^2)\tr)\). Fix disjoint, effective
positive-integer codes for signs, matrix digits, and terminal records.
If the rule runs forever, the nested triangles recover \(x\), and
the sign and \(x_2/x_1\) recover \(\alpha\).

The initial direction is cubic exactly when \(b\) has degree three.
For degree three, \(1,b,b^2\) form a basis of \(\mathbb Q(b)\).
Conversely, if these three elements form a basis of a cubic field
\(K\), then \(b\) generates \(K\) because its powers span \(K\).
Proposition~\ref{prop:global-rule} now gives a constant matrix tail in
the cubic case and unbounded entries in every other infinite case.
An injective code for matrices preserves the distinction, since an
eventually periodic sequence uses only finitely many digits.

If a rational relation \(c x^{(n)}=0\) is detected after the prefix
product \(Q_n\), then \(b\) is a root of the polynomial
\[
 p(T)=cQ_n^{-1}(1,T,T^2)\tr .
\]
The polynomial is nonzero because \(c\ne0\) and \(Q_n\) is invertible.
Encode \(p\) and the index of \(b\) among its distinct real roots as
a terminal record. The root index is obtained by comparisons with
rational separators between the roots. Append the sequence
\(1,2,3,\ldots\) after the terminal record. Use a dedicated terminal
record for zero with the same aperiodic tail. This makes the expansion
infinite and recoverable on all inputs, including those that halt.

At any step, the sign of a rational linear form on \(x^{(n)}\) is
the sign of the corresponding rational linear form on
\(Q_n^{-1}(1,b,b^2)\tr\), since the normalization factor is positive.
After fixing the sign of \(\alpha\), these are polynomial comparisons
of degree at most two in \(\alpha\). Each stage and each digit has
already been shown to require only finitely many operations.

\begin{samepage}
Finally, let \(Q\) be the prefix before the constant tail and let \(P\)
be its repeating matrix. If \(w\) is the positive Perron eigenvector
of \(P\), then the recovered direction is \(\pi(Qw)\). Both \(P\)
and \(P^{-1}\) are integral, so the Perron eigenvalue \(\lambda_P\)
and its inverse are algebraic integers. The direction \(w\) can be
represented over \(\mathbb Q(\lambda_P)\), and consequently
\(\alpha\in\mathbb Q(\lambda_P)\). Since \(\alpha\) has degree three
and \(P\) is \(3\times3\), this gives
\(\mathbb Q(\lambda_P)=\mathbb Q(\alpha)\).\qedhere
\par
\end{samepage}
\end{proof}

\section*{Acknowledgment of AI assistance}

OpenAI Codex assisted in developing the construction and proof and in
preparing the manuscript.

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\raggedright

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\bibitem{tagnon}
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\end{thebibliography}

\end{document}