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\begin{document}
\vspace*{2cm}

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\centerline{\Large\textbf{Normaliz \version}}

\vspace*{1.5cm}

\begin{center}Winfried Bruns\qquad Max Horn\\[14pt]
	Team member for fusion rings: Sébastien Palcoux\\[14pt]
	Former Normaliz~3 team members: Tim R\"omer, Richard Sieg,\\ Christof S\"oger and Ulrich von der Ohe\\[14pt]
	Normaliz~2 team member: Bogdan Ichim\\[14pt]
	\url{https://normaliz.uos.de}\qquad\qquad\qquad
	\url{https://github.com/Normaliz}\\[14pt]
	\url{mailto:normaliz@uos.de}\\[14pt]
	\url{https://hub.docker.com/r/normaliz/normaliz/}\\[14pt]
	\url{https://mybinder.org/v2/gh/Normaliz/NormalizJupyter/master}\\[14pt]
	Short reference: \verb|NmzShortRef.pdf|
\end{center}



\tableofcontents
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  INTRODUCTION  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\include{Intro}
\include{Discrete}
\include{Monoid}
\include{Input}
\include{Options}
\include{Advanced}
\include{Algebraic}
\include{OptFiles}
\include{Technical}
\include{Appendix}
\newpage

\addcontentsline{toc}{section}{References}
\begin{thebibliography}{15.}
\small

\bibitem{4ti2} 
4ti2 team. 4ti2-A software package
for algebraic, geometric and combinatorial problems on linear
spaces. Available at \url{https://github.com/4ti2/4ti2}.

\bibitem{CoCoA}
J.~Abbott, A.~M.~Bigatti and G.~Lagorio,
\emph{CoCoA-5: a system for doing Computations in Commutative Algebra}.
Available at \url{http://cocoa.dima.unige.it}.

\bibitem{ABPP}
M.A. Alekseyev, W. Bruns, S. Palcoux and F. V. Petrov, \emph{Classification of integral modular data up to rank 13}. Preprint \url{arXiv:2302.01613}.

\bibitem{AI}
V.~Almendra and B.~Ichim,
\emph{jNormaliz~1.7}.
Available at \url{https://normaliz.uos.de}.

\bibitem{LatInt}
V.~Baldoni, N.~Berline, J.~A.~De~Loera, B.~Dutra,
M.~K\"oppe, S.~Moreinis, G.~Pinto, M.~Vergne and J.~Wu,
\emph{A User's Guide for LattE integrale~v1.7.2, 2013}.
Software package LattE is available at \url{https://www.math.ucdavis.edu/~latte/}.

\bibitem{Bremner}
D.~Bremner, M.~D.~Sikiri\'c, D.~V.~Pasechnik, Th.~Rehn and A.~Sch\"urmann,
\emph{Computing symmetry groups of polyhedra}.
LMS J.\ Comp.\ Math.\ 17 (2014), 565--581.

\bibitem{has}
St.~ Brumme,
\emph{Hash library}.
Package available at \url{https://create.stephan-brumme.com/}.

\bibitem{BruAuto}
W. Bruns, \emph{Automorphism groups and normal forms in Normaliz.} Res. Math. Sci. 9 (2022), no. 2, Paper No. 20, 15 pp.

\bibitem{BruVol}
W. Bruns, \emph{Polytope volume in Normaliz.}
São Paulo J. Math. Sci.
\url{https://doi.org/10.1007/s40863-022-00317-9}

\bibitem{BGOW}
W.~Bruns, P.~Garcia-Sanchez, C.~O'Neill and D.~Wilburne,
\emph{Wilf's conjecture in fixed multiplicity}.
% Preprint \url{arXiv:1903.04342}.
Int.\ J.\ Algebra Comp.\ 30 (2020), 861--882.

\bibitem{BG}
W.~Bruns and J.~Gubeladze,
\emph{Polytopes, rings, and K-theory}.
Springer, 2009.

\bibitem{BHIKS}
W.~Bruns, R.~Hemmecke, B.~Ichim, M.~K\"oppe and C.~S\"oger,
\emph{Challenging computations of Hilbert bases of cones associated with algebraic statistics}.
Exp.\ Math.\ 20 (2011), 25--33.

\bibitem{BI}
W.~Bruns and B.~Ichim,
\emph{Normaliz: algorithms for rational cones and affine monoids}.
J.\ Algebra 324 (2010) 1098--1113.

\bibitem{BI2}
W.~Bruns and B.~Ichim,
\emph{Polytope volume by descent in the face lattice and applications in social choice}.
% Preprint \url{arXiv:1807.02835}.
Math.\ Prog.\ Comp. 113 (2020), 415--442.

\bibitem{BIS}
W.~Bruns, B.~Ichim and C.~S\"oger,
\emph{The power of pyramid decomposition in Normaliz}.
J.\ Symb.\ Comp.\ 74 (2016), 513--536.

\bibitem{BIS2}
W.~Bruns, B.~Ichim and C.~S\"oger,
\emph{Computations of volumes and Ehrhart series in four candidates elections}.
Ann.\ Oper.\ Res.\ 280 (2019), 241--265.

\bibitem{BK02}
W.~Bruns and R.~Koch,
\emph{Computing the integral closure of an affine semigroup}.
Univ.\ Iagell.\ Acta Math.\ 39 (2001), 59--70.

\bibitem{BSS}
W.~Bruns, R.~Sieg and C.~S\"oger,
\emph{Normaliz~2013--2016}.
% To appear in the final report of the DFG~SPP~1489.
% Preprint \url{arXiv:1611.07965}.
In
G.~B\"ockle, W.~Decker and G.~Malle, editors,
\emph{Algorithmic and Experimental Methods in Algebra, Geometry, and Number Theory},
pages~123--146.
Springer, 2018.

\bibitem{BS}
W.~Bruns and C.~S\"oger,
\emph{The computation of weighted Ehrhart series in Normaliz}.
J.\ Symb.\ Comp.\ 68 (2015), 75--86.

\bibitem{vinci} B. B\"{u}eler and A. Enge, \emph{Vinci.} Package available from \url{https://www.math.u-bordeaux.fr/~aenge/}

\bibitem{practical}	
B. Büeler, A. Enge, K. Fukuda, \emph{Exact volume computation for polytopes: a practical study.} In: Polytopes - combinatorics and computation (Oberwolfach, 1997), pp. 131 -- 154,
DMV Sem. 29, Birkhäuser, Basel, 2000.

\bibitem{DLHK}	
J. A. De Loera, R. Hemmecke and M. Köppe.
Algebraic and geometric ideas in the theory of discrete optimization.
MOS-SIAM Series on Optimization, 14. Society for Industrial and Applied Mathematics (SIAM), Philadelphia 2013.

\bibitem{e-antic}
V.~Delecroix,
\emph{embedded algebraic number fields (on top of antic)},
package available at \url{https://github.com/flatsurf/e-antic}.

\bibitem{DS}
J.Dong and A. Schopieray, \emph{Near-integral fusion}. Preprint \url{arXiv:2407.15955}.

\bibitem{EGNO}
P.~Etingof, S.~Gelaki, D.~Nikshych, and V.~Ostrik, {\em Tensor Categories}, American Mathematical Society, (2015).

\bibitem{Filli} 
P. Filliman, \emph{The volume of duals and sections of polytopes. } Mathematika 37 (1992), 67--80.

\bibitem{GAP-NmzInterface}
S.~Gutsche, M.~Horn and C.~S\"oger,
\emph{NormalizInterface for GAP}.
Available at \url{https://github.com/gap-packages/NormalizInterface}.

\bibitem{PyNormaliz}
S.~Gutsche and R.~Sieg,
\emph{PyNormaliz - an interface to Normaliz from python}.
Available at \url{https://github.com/Normaliz/PyNormaliz}.

\bibitem{Flint}
W.~B.~Hart, F.~Johansson and S.~Pancratz,
\emph{FLINT: Fast Library for Number Theory}.
Available at \url{https://flintlib.org}.

\bibitem{HM}
Hemmecke and P. N. Malkin. Computing generating sets of lattice ideals and Markov bases of
lattices. J. Symb. Comp. 44, 1463--1476 (2009).

\bibitem{Lawrence} J. Lawrence, \emph{Polytope volume computation.} Math. Comp. 57 (1991), 259--271.

\bibitem{KV}
M.~K\"oppe and S.~Verdoolaege,
\emph{Computing parametric rational generating functions with a primal Barvinok algorithm}.
Electron.\ J.\ Comb.\ 15, No.\ 1, Research Paper~R16, 19~p.\ (2008).

\bibitem{nauty}
B.~D.~McKay and A.~Piperno,
\emph{Practical graph isomorphism,~II}.
J.\ Symbolic Comput.\ 60 (2014), 94--112.

\bibitem{Ost}
V. Ostrik, \emph{Pivotal fusion categories of rank 3.} Mosc. Math. J. 15 (2015), no. 2, 373--396, 405.

\bibitem{Po}
L.~Pottier,
\emph{The Euclide algorithm in dimension~$n$}.
Research report, ISSAC~96, ACM Press 1996.

\bibitem{Sch}
A.~Sch\"urmann,
\emph{Exploiting polyhedral symmetries in social choice}.
Social Choice and Welfare 40 (2013), 1097--1110.

\bibitem{Stu}
B. Sturmfels,
\emph{Gröbner baes and convex polytopes}.
American Mathematical Society 1996.
\end{thebibliography}
\newpage

\addcontentsline{toc}{section}{\indexname}
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\end{document}

Relations do not select a
sublattice of $\ZZ^d$ or a subcone of $\RR^d$, but define a
monoid as a quotient of $\ZZ_+^d$ modulo a system of
congruences (in the semigroup sense!).

The rows of the input matrix of this type are interpreted as
generators of a subgroup $U\subset\ZZ^d$, and Normaliz computes an affine monoid and its normalization as explained in Section~\ref{binomials}.

Set $G=\ZZ^d/U$ and $L=G/\textup{torsion}(G)$. Then the ambient lattice
is $\AA=\ZZ^r$, $r=\rank L$, and the efficient lattice is $L$, realized
as a sublattice of $\AA$. Normaliz computes the image $M$ of $\ZZ^d_+$ in $L$ and its normalization. To this end, $M$ is embedded into a lattice $\ZZ$, $r=\rank M$. There is no canonical choice for such an mebdding, but if possible, Normaliz finds an embedding into $\ZZ_+^r$.