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\ihead{Errata to ``On finite sequences satisfying linear recursions''}
\ohead{\today}
\begin{document}

\begin{center}
\textbf{On finite sequences satisfying linear recursions}

\textit{Noam D. Elkies}

arXiv:math/0105007v2 ( \url{https://arxiv.org/abs/math/0105007v2} )

version of 12 June 2002 (arXiv version v2)

\textbf{Errata and addenda by Darij Grinberg}

\bigskip
\end{center}

%\setcounter{section}{10}


\section*{Errata}

The numbering of pages in the following errata matches the numbering of pages
in the arXiv version (v2).

The errata cover everything until after the proof of the Corollary on page 8.

\begin{itemize}
\item \textbf{page 1, Abstract:} In the first sentence, the subsentence
\textquotedblleft subset of sequences satisfying a degree-$m$ linear
recursion, i.e. for which there exist $a_{0},\ldots,_{m}\in k$%
\textquotedblright\ is repeated twice. (Only in the actual PDF; not in the
arXiv metadata abstract.)

\item \textbf{page 2:} You write: \textquotedblleft linear subspaces of
dimensions $m$ and $m+1$ in $W_{n}$\textquotedblright. I think the
\textquotedblleft$m+1$\textquotedblright\ should be an \textquotedblleft%
$m-1$\textquotedblright\ here. The same applies to the abstract (and this one
also appears in the arXiv metadata abstract).

\item \textbf{page 3:} \textquotedblleft Let $W_{n}$ as the
dual\textquotedblright\ $\rightarrow$ \textquotedblleft Let $W_{n}$ be the
dual\textquotedblright.

\item \textbf{page 3, Lemma 1:} I would replace the word \textquotedblleft
adjoint\textquotedblright\ by \textquotedblleft dual map\textquotedblright,
since the word \textquotedblleft adjoint\textquotedblright\ has too many
different meanings.

\item \textbf{page 3, proof of Lemma 1:} In the first displayed equation of
this proof, an \textquotedblleft equals\textquotedblright\ sign is missing;
i.e., the equation should be
\[
P\left(  Y,Z\right)  =\sum_{j=0}^{n-m}b_{j}Y^{j}Z^{n-m-j}.
\]


\item \textbf{page 4, The ideal }$I_{x}$\textbf{:} It may be helpful to
mention that $I_{x}$ is the annihilator of $x$, where we let the graded
$k$-algebra $k\left[  Y,Z\right]  =\bigoplus\limits_{n\geq0}V_{n}$ act on its
graded dual $\bigoplus\limits_{n\geq0}V_{n}^{\ast}=\bigoplus\limits_{n\geq
0}W_{n}$ in the obvious way (i.e., any $Q\in k\left[  Y,Z\right]  $ acts on
$x\in\bigoplus\limits_{n\geq0}V_{n}^{\ast}$ by pre-composition). This explains
why $I_{x}$ is a homogeneous ideal, provided that one knows the basic
properties of graded algebras.

\item \textbf{page 4, Proposition 1:} The \textquotedblleft$V_{m-m_{0}}%
$\textquotedblright\ on the right hand side of (11) needs to be understood as
$0$ if $m-m_{0}<0$. In view of this, it may be worth (at least in Proposition
1) defining $V_{i}=W_{i}=0$ for all $i<0$, not just for $i=-1$.

\item \textbf{page 5, Lemma 4:} Here you are using without mention and proof
the following facts about bivariate polynomials:

\begin{enumerate}
\item Any two nonzero polynomials $Q_{1},Q_{2}\in k\left[  Y,Z\right]  $ have
a gcd and an lcm. These are unique up to multiplication by a nonzero scalar,
and can be chosen to satisfy $\gcd\left(  Q_{1},Q_{2}\right)  \cdot
\operatorname{lcm}\left(  Q_{1},Q_{2}\right)  =Q_{1}Q_{2}$.

\item If $Q_{1}$ and $Q_{2}$ are nonzero homogeneous polynomials in $k\left[
Y,Z\right]  $, then $\gcd\left(  Q_{1},Q_{2}\right)  $ and $\operatorname{lcm}%
\left(  Q_{1},Q_{2}\right)  $ are homogeneous polynomials as well.
\end{enumerate}

It took me a while to figure out why these facts are true. In hindsight, it
was pretty easy: Fact 1 holds in any unique factorization domain (and
$k\left[  Y,Z\right]  $ is known to be a unique factorization domain); Fact 2
follows from the (easily provable) fact that any polynomial that divides a
nonzero homogeneous polynomial in $k\left[  Y,Z\right]  $ is itself
homogeneous. Alternatively, Facts 1 and 2 can be proved together rather nicely
as follows: Define a map $\Phi$ from%
\[
\left\{  \text{nonzero homogeneous polynomials in }k\left[  Y,Z\right]
\right\}
\]
to%
\[
\mathbb{N}\times\left\{  \text{homogeneous polynomials in }k\left[  X\right]
\right\}
\]
which sends each nonzero homogeneous polynomial $P\in k\left[  Y,Z\right]  $
to the pair $\left(  v_{Y}\left(  P\right)  ,P\left(  1,X\right)  \right)  $,
where $v_{Y}\left(  P\right)  $ is the largest $i\in\mathbb{N}$ satisfying
$Y^{i}\mid P$. This map $\Phi$ is easily seen to be a monoid isomorphism from%
\[
\left(  \left\{  \text{nonzero homogeneous polynomials in }k\left[
Y,Z\right]  \right\}  ,\cdot\right)
\]
to%
\[
\left(  \mathbb{N},+\right)  \times\left(  \left\{  \text{homogeneous
polynomials in }k\left[  X\right]  \right\}  ,\cdot\right)  ;
\]
thus, it induces an isomorphism between the divisibility posets of these two
monoids. Since the divisibility poset of the latter monoid is a distributive
lattice (because it is a direct product of two distributive lattices), it thus
follows that so is the divisibility poset of the former poset. But this means
that nonzero homogeneous polynomials in $k\left[  Y,Z\right]  $ have gcds and
lcms in the set of all nonzero homogeneous polynomials in $k\left[
Y,Z\right]  $. It is not hard to check that these gcds and lcms are also gcds
and lcms in the entire polynomial ring $k\left[  Y,Z\right]  $. Thus, Facts 1
and 2 easily follow.

This was a fun exercise, but I feel that some readers might prefer a reference.

\item \textbf{page 6, proof of Proposition 1:} After \textquotedblleft We can
now easily prove Prop. 1.\textquotedblright, I would add: \textquotedblleft
There exists a nonzero $Q\in I_{x}\cap V_{m_{0}}$, since $I_{x}$ is a
homogeneous ideal.\textquotedblright\ (Otherwise, per se, we only know that
there is a nonzero polynomial $Q\in I_{x}$ of degree $m_{0}$, not necessarily
a homogeneous one.)

\item \textbf{page 6, proof of Proposition 1:} Replace \textquotedblleft and
$Q\in I_{x}\cap V_{m_{0}}-\left\{  0\right\}  $ then\textquotedblright\ by
\textquotedblleft and $Q\in I_{x}\cap V_{m}-\left\{  0\right\}  $
then\textquotedblright.

\item \textbf{page 6, proof of Proposition 1:} Replace \textquotedblleft thus
$I_{x}\cap V_{m_{0}}$ consists\textquotedblright\ by \textquotedblleft thus
$I_{x}\cap V_{m}$ consists\textquotedblright.

\item \textbf{page 8:} After proving the Corollary, you write
\textquotedblleft has only $m$ rows\textquotedblright. I think this should be
\textquotedblleft has only $m$ columns\textquotedblright.

\item \textbf{page 9, proof of Lemma 5:} Replace \textquotedblleft%
$\widehat{\chi}_{K}\left(  y\right)  $\textquotedblright\ by \textquotedblleft%
$\widehat{\chi_{K}}\left(  a\right)  $\textquotedblright.

\item \textbf{general notations:} You are using the $\geq$ (%
%TCIMACRO{\TEXTsymbol{\backslash}}%
%BeginExpansion
$\backslash$%
%EndExpansion
geq) and $\geqslant$ (%
%TCIMACRO{\TEXTsymbol{\backslash}}%
%BeginExpansion
$\backslash$%
%EndExpansion
geqslant) symbols interchangeably. It is probably best to replace each
\textquotedblleft$\geq$\textquotedblright\ by \textquotedblleft$\geqslant
$\textquotedblright\ for the sake of consistency with \textquotedblleft%
$\leqslant$\textquotedblright\ (which you use consistently).
\end{itemize}


\end{document}