## Normaliz

 swMATH ID: 630 Software Authors: Winfried Bruns, Bogdan Ichim, Tim Römer, Christof Söger Description: Normaliz is a tool for computations in affine monoids, vector configurations, lattice polytopes, and rational cones. Its input data can be specified in terms of a system of generators or vertices or a system of linear homogeneous Diophantine equations, inequalities and congruences or a binomial ideal. Normaliz computes the dual cone of a rational cone (in other words, given generators, Normaliz computes the defining hyperplanes, and vice versa), convex hulls, a triangulation of a vector, the Hilbert basis of a (not necessarily pointed) rational cone, the lattice points of a rational polytope or unbounded polyhedron, the integer hull, the normalization of an affine monoid, the Hilbert (or Ehrhart) series and the Hilbert (or Ehrhart) (quasi) polynomial under a Z-grading (for example, for rational polytopes), generalized (or weighted) Ehrhart series and Lebesgue integrals of polynomials over rational polytopes via NmzIntegrate, a description of the cone and lattice under consideration by a system of inequalities, equations and congruences. Homepage: https://www.normaliz.uni-osnabrueck.de/ Programming Languages: C,C++ Dependencies: Note that it may be necessary to re-compile Normaliz on your system. Normaliz is compatible with the GNU C++ compiler (source included in the main package) and Microsoft Visual Studio configured for the Intel� C++ compiler. Keywords: Convex hull; affine monoid; Hilbert basis; normalization; Ehrhart series; lattice points; rational polytope Related Software: Macaulay2; 4ti2; polymake; SINGULAR; CoCoA; SageMath; GAP; LattE; cdd; Help; GitHub; SCIP; Gfan; lrs; nauty; PORTA; TOPCOM; NormalizInterface; numericalsgps; CoCoALib Referenced in: 176 Publications This software is also referenced in ORMS.
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### Referenced by 235 Authors

 21 Bruns, Winfried 18 Villarreal, Rafael Heraclio 9 Söger, Christof 8 Ichim, Bogdan 8 Paffenholz, Andreas 7 García Sánchez, Pedro A. 5 Gitler, Isidoro 5 Nill, Benjamin 5 Sturmfels, Bernd 4 Bächle, Andreas 4 García García, Juan Ignacio 4 Hibi, Takayuki 4 Marin-Aragon, Daniel 4 Michałek, Mateusz 4 Schürmann, Achill 4 Simis, Aron 4 Thomas, Rekha R. 4 Valencia, Carlos E. 4 Yoshida, Ruriko 3 Abbott, John A. 3 Bigatti, Anna Maria 3 Gubeladze, Joseph 3 Haase, Christian Alexander 3 Hemmecke, Raymond 3 Herr, Katrin 3 Joswig, Michael 3 Köppe, Matthias 3 Lorenz, Benjamin 3 Margolis, Leo 3 Martínez-Bernal, José 3 Montaño, Jonathan 3 Moyano Fernández, Julio José 3 Ohsugi, Hidefumi 3 Olive, Marc 3 Rehn, Thomas 3 Schepers, Jan 3 Sullivant, Seth 3 Vigneron-Tenorio, Alberto 2 Al-Ayyoub, Ibrahim 2 Assi, Abdallah 2 Beck, Matthias 2 Boffi, Giandomenico 2 Bogart, Tristram 2 Braun, Benjamin 2 Conca, Aldo 2 Diss, Mostapha 2 Escobar, Cesar A. 2 Herzog, Jürgen 2 Kappl, Rolf 2 Katthän, Lukas 2 Keicher, Simon 2 Kimmerle, Wolfgang 2 Koch, Robert 2 Kohl, Florian 2 Konovalov, Olexandr 2 Kubjas, Kaie 2 Logar, Alessandro 2 Olsen, McCabe 2 O’Neill, Christopher 2 Rauh, Johannes 2 Ren, Yue 2 Reyes, Enrique 2 Rossmann, Tobias 2 Santos, Francisco 2 Sieg, Richard 2 Swanson, Irena 2 Takemura, Akimichi 2 Thiel, Marko 2 Van Langenhoven, Leen 2 Vasconcelos, Wolmer V. 1 Al-Zoubi, Khaldoun 1 Arkani-Hamed, Nima 1 Assarf, Benjamin 1 Auffray, Nicolas 1 Avis, David M. 1 Ba, Thang Luu 1 Binh, Hong Ngoc 1 Bivià-Ausina, Carles 1 Böhm, Janko 1 Bonanzinga, Vittoria 1 Borowka, Sophia 1 Borzì, Alessio 1 Brenner, Holger 1 Breuer, Felix 1 Bulger, David W. 1 Burr, Michael A. 1 Burton, Benjamin A. 1 Caicedo, Mauricio 1 Castillo, Federico 1 Colmenarejo, Laura 1 Craw, Alastair 1 D’Alì, Alessio 1 D’Anna, Marco 1 Dao, Hailong 1 De Loera, Jesús A. 1 Decker, Wolfram 1 del Campo, Abraham Martín 1 Delgado, Manuel 1 Deshpande, Tanmay 1 Díaz-Ramírez, J. D. ...and 135 more Authors
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### Referenced in 78 Serials

 10 Journal of Symbolic Computation 9 Journal of Algebra 7 International Journal of Algebra and Computation 6 Journal of Pure and Applied Algebra 6 Discrete & Computational Geometry 4 Discrete Mathematics 4 Mathematical Programming Computation 3 Beiträge zur Algebra und Geometrie 3 Advances in Mathematics 3 Proceedings of the American Mathematical Society 3 European Journal of Combinatorics 3 Advances in Applied Mathematics 3 Journal of Algebraic Combinatorics 3 Annals of Combinatorics 3 Journal of High Energy Physics 3 ACM Communications in Computer Algebra 3 The Journal of Software for Algebra and Geometry 2 Discrete Applied Mathematics 2 Mathematics of Computation 2 Annals of the Institute of Statistical Mathematics 2 Nagoya Mathematical Journal 2 Results in Mathematics 2 Theory and Decision 2 Tôhoku Mathematical Journal. Second Series 2 Experimental Mathematics 2 Journal of Algebra and its Applications 2 Research in the Mathematical Sciences 2 RSME Springer Series 1 Archive for Rational Mechanics and Analysis 1 Computer Physics Communications 1 Indian Journal of Pure & Applied Mathematics 1 Israel Journal of Mathematics 1 Nuclear Physics. B 1 Rocky Mountain Journal of Mathematics 1 Acta Mathematica Vietnamica 1 Archiv der Mathematik 1 Automatica 1 Collectanea Mathematica 1 Illinois Journal of Mathematics 1 Inventiones Mathematicae 1 Journal of Combinatorial Theory. Series A 1 Kodai Mathematical Journal 1 Mathematische Annalen 1 Memoirs of the American Mathematical Society 1 Michigan Mathematical Journal 1 Proceedings of the London Mathematical Society. Third Series 1 Ricerche di Matematica 1 Semigroup Forum 1 Theoretical Computer Science 1 Transactions of the American Mathematical Society 1 Mathematical Social Sciences 1 Operations Research Letters 1 Social Choice and Welfare 1 Graphs and Combinatorics 1 SIAM Journal on Discrete Mathematics 1 SIAM Journal on Matrix Analysis and Applications 1 Annals of Operations Research 1 Linear Algebra and its Applications 1 Mathematical Programming. Series A. Series B 1 Journal of Algebraic Geometry 1 Combinatorics, Probability and Computing 1 Zeszyty Naukowe Uniwersytetu Jagiellońskiego. Universitatis Iagellonicae Acta Mathematica 1 The Electronic Journal of Combinatorics 1 Analele Științifice ale Universității “Ovidius” Constanța. Seria: Matematică 1 The Ramanujan Journal 1 Journal of Combinatorial Optimization 1 Revista Matemática Complutense 1 Journal of Integer Sequences 1 Foundations of Computational Mathematics 1 Mediterranean Journal of Mathematics 1 Discrete Optimization 1 London Mathematical Society Lecture Note Series 1 Journal of Mathematics and Music 1 Optimization Letters 1 Algebra & Number Theory 1 European Journal of Mathematics 1 Springer Monographs in Mathematics 1 Discrete Mathematics and its Applications
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### Referenced in 27 Fields

 93 Commutative algebra (13-XX) 58 Convex and discrete geometry (52-XX) 51 Algebraic geometry (14-XX) 43 Combinatorics (05-XX) 30 Group theory and generalizations (20-XX) 24 Computer science (68-XX) 19 Number theory (11-XX) 17 Operations research, mathematical programming (90-XX) 10 Associative rings and algebras (16-XX) 9 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 7 Statistics (62-XX) 6 Order, lattices, ordered algebraic structures (06-XX) 6 Linear and multilinear algebra; matrix theory (15-XX) 5 Quantum theory (81-XX) 3 History and biography (01-XX) 3 Manifolds and cell complexes (57-XX) 3 Numerical analysis (65-XX) 3 Biology and other natural sciences (92-XX) 2 Several complex variables and analytic spaces (32-XX) 2 Geometry (51-XX) 2 Differential geometry (53-XX) 2 Probability theory and stochastic processes (60-XX) 1 General and overarching topics; collections (00-XX) 1 $$K$$-theory (19-XX) 1 Mechanics of particles and systems (70-XX) 1 Mechanics of deformable solids (74-XX) 1 Systems theory; control (93-XX)