% Hexagon version of critical-values.tex (based on the arXiv submission).
%
% This submission uses the same pdfLaTeX setup as the arXiv version.
% It replaces the fontspec/unicode-math setup of the original with a
% pdfLaTeX one and carries no system fonts. Palatino text with Euler math keeps
% the original's colour; the body below is otherwise identical.
%
% The bibliography is inlined as thebibliography rather than left to
% biblatex+biber, so no .bbl has to be shipped or version-matched.
%
% Build: pdflatex critical-values-hexagon (twice, for the cross-references).
\documentclass[10pt]{article}

\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{newpxtext}    % Palatino-like text, in place of Miller Display
\usepackage{amsmath}
\usepackage{newpxmath}    % Palatino-matched italic math
% Not eulervm: the original loads it but unicode-math overrides it, so the
% XeLaTeX build actually sets math in Latin Modern Math -- italic letters.
% Euler here would set them upright and look nothing like the original.

\usepackage{geometry}
\geometry{letterpaper, margin=1in}
\renewcommand{\baselinestretch}{1.2}

\usepackage{titlesec}
\titleformat{\section}{\normalfont\large\bfseries}{\thesection}{1em}{}
\titleformat{\subsection}{\normalfont\normalsize\bfseries}{\thesubsection}{1em}{}

\usepackage{url}
\usepackage{hyperref}
\hypersetup{
  pdfauthor={RJ Acu\~na},
  pdftitle={Every Algebraic Number is a Critical Value of an Integer Polynomial},
  pdflang=en-US,
  colorlinks=true,
  urlcolor = blue,
  linkcolor = blue,
  citecolor = blue}

\newcommand\Z{\mathbb{Z}}
\newcommand\Q{\mathbb{Q}}

\usepackage{graphicx}
\usepackage[normalem]{ulem}
\usepackage[makeroom]{cancel}

% newpxmath defines \openbox and so does amsthm; let the latter win.
\let\openbox\relax
\usepackage{amsthm}
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem*{theorem*}{Theorem}
\newtheorem*{ptheorem}{Petrov's Theorem}
\newtheorem{cor}{Corollary}
\newtheorem{prop}{Proposition}
\theoremstyle{remark}
\newtheorem{example}{Example}
\theoremstyle{plain}

% Proofs open with a wavy-underlined marker and close with a black box.
\newenvironment{pf}{%
  \par\addvspace{\medskipamount}\noindent\uwave{pf.}\ \ignorespaces}{%
  \unskip\nobreak\hfill$\blacksquare$\par\addvspace{\medskipamount}}

\usepackage{microtype}
\microtypesetup{protrusion=false}
\usepackage{nopageno}
\usepackage{ragged2e}


% Shared-body hooks (see body.tex).
\newif\ifanonymous
\anonymousfalse
\newcommand\paperSection[1]{\section{#1}}
\newcommand\secref[1]{\S\ref{#1}}

\begin{document}

\begin{center}
  {\Large \textsc Every Algebraic Number is a
    Critical Value of an Integer Polynomial}

  {by RJ Acu\~na}
\end{center}

\begin{center}
  \begin{minipage}{0.85\textwidth}
    \begin{abstract}
      \noindent
      \justifying
      \input{abstract}
    \end{abstract}
  \end{minipage}
\end{center}

\input{body}

\begin{thebibliography}{Acu26b}

\bibitem[Ded78]{dedekind1878}
Richard Dedekind.
\newblock \"{U}ber den Zusammenhang zwischen der Theorie der Ideale und der
  Theorie der h\"{o}heren Kongruenzen.
\newblock \emph{Abhandlungen der K\"{o}niglichen Gesellschaft der
  Wissenschaften zu G\"{o}ttingen}, 1878.

\bibitem[SVW17]{svw2017}
Albert Schwarz, Vadim Vologodsky, and Johannes Walcher.
\newblock \emph{Integrality of Framing and Geometric Origin of 2-functions}.
\newblock \url{https://arxiv.org/abs/1702.07135}, 2017.

\bibitem[The20]{mathlib}
The mathlib Community.
\newblock The {L}ean Mathematical Library.
\newblock In \emph{Certified Programs and Proofs (CPP 2020)}. ACM, 2020.

\bibitem[MU21]{lean4}
Leonardo de Moura and Sebastian Ullrich.
\newblock The {L}ean 4 Theorem Prover and Programming Language.
\newblock In \emph{Automated Deduction -- CADE 28}, Lecture Notes in Computer
  Science. Springer, 2021.

\bibitem[Pet21]{petrov2021}
Fedor Petrov.
\newblock Answer to ``When $p(x)^{2} \mid f(g(x))$?''.
\newblock \url{https://mathoverflow.net/q/397800}, 2021.

\bibitem[Acu26a]{app}
RJ Acu\~{n}a.
\newblock \emph{Critical values of integer polynomials: an online calculator}.
\newblock \url{https://rjacuna.github.io/CriticalValues/}, 2026.

\bibitem[Acu26b]{repo}
RJ Acu\~{n}a.
\newblock \emph{{CriticalValues}: paper, {L}ean~4 formalization and calculator}.
\newblock \url{https://github.com/rjacuna/CriticalValues}, 2026.

\end{thebibliography}

\begin{center}
  \begin{tabular}{@{}c@{\hspace{3em}}c@{}}
    \includegraphics[width=0.2\textwidth]{qr-repo.png} &
    \includegraphics[width=0.2\textwidth]{qr-site.png} \\[4pt]
    \small the repository & \small the calculator
  \end{tabular}
\end{center}

\end{document}