{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,27]],"date-time":"2026-02-27T04:33:01Z","timestamp":1772166781089,"version":"3.50.1"},"reference-count":42,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2018,9,5]],"date-time":"2018-09-05T00:00:00Z","timestamp":1536105600000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Found Comput Math"],"published-print":{"date-parts":[[2019,8]]},"DOI":"10.1007\/s10208-018-9396-x","type":"journal-article","created":{"date-parts":[[2018,9,5]],"date-time":"2018-09-05T00:20:54Z","timestamp":1536106854000},"page":"775-812","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":14,"title":["The Geometry of Gaussoids"],"prefix":"10.1007","volume":"19","author":[{"given":"Tobias","family":"Boege","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Alessio","family":"D\u2019Al\u00ec","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Thomas","family":"Kahle","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Bernd","family":"Sturmfels","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2018,9,5]]},"reference":[{"key":"9396_CR1","doi-asserted-by":"publisher","first-page":"815","DOI":"10.4171\/JEMS\/680","volume":"19","author":"F Ardila","year":"2017","unstructured":"F.\u00a0Ardila, F.\u00a0Rinc\u00f3n and L.\u00a0Williams: Positively oriented matroids are realizable, J.\u00a0Eur.\u00a0Math.\u00a0Soc. (JEMS) 19 (2017) 815\u2013833.","journal-title":"J. Eur. Math. Soc. (JEMS)"},{"key":"9396_CR2","unstructured":"M.\u00a0Baker and N.\u00a0Bowler: Matroids over partial hyperstructures, \n                    arXiv:1709.09707\n                    \n                  ."},{"key":"9396_CR3","unstructured":"A.\u00a0Bj\u00f6rner, M.\u00a0Las Vergnas, B.\u00a0Sturmfels, N.\u00a0White and G.\u00a0Ziegler: Oriented Matroids, Cambridge University Press, 1993."},{"key":"9396_CR4","doi-asserted-by":"publisher","first-page":"21","DOI":"10.1016\/S0195-6698(13)80052-2","volume":"11","author":"J Bokowski","year":"1990","unstructured":"J.\u00a0Bokowski and J.\u00a0Richter: On the finding of final polynomials, European J.\u00a0Combinatorics 11 (1990) 21\u201334.","journal-title":"European J. Combinatorics"},{"key":"9396_CR5","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-2066-4","volume-title":"Coxeter Matroids","author":"A Borovik","year":"2003","unstructured":"A.\u00a0Borovik, I.M.\u00a0Gelfand and N.\u00a0White: Coxeter Matroids, Progress in Mathematics 216, Birkh\u00e4user, Boston, MA, 2003."},{"key":"9396_CR6","doi-asserted-by":"publisher","first-page":"25","DOI":"10.1007\/s13366-016-0316-4","volume":"58","author":"S Brodsky","year":"2017","unstructured":"S.\u00a0Brodsky, C.\u00a0Ceballos and J-P.\u00a0Labb\u00e9: Cluster algebras of type D4, tropical planes, and the positive tropical Grassmannian, Beitr.\u00a0Algebra Geom.\u00a058 (2017) 25\u201346.","journal-title":"Beitr. Algebra Geom."},{"key":"9396_CR7","doi-asserted-by":"publisher","first-page":"33","DOI":"10.1016\/0893-9659(90)90009-Z","volume":"3","author":"A Dress","year":"1990","unstructured":"A.\u00a0Dress and W.\u00a0Wenzel: Valuated matroids: a new look at the greedy algorithm, Appl.\u00a0Math.\u00a0Letters 3 (1990) 33\u201335.","journal-title":"Appl. Math. Letters"},{"key":"9396_CR8","doi-asserted-by":"publisher","first-page":"68","DOI":"10.1016\/0001-8708(91)90036-7","volume":"86","author":"A Dress","year":"1991","unstructured":"A.\u00a0Dress and W.\u00a0Wenzel: Grassmann\u2013Pl\u00fccker relations and matroids with coefficients, Adv.\u00a0Math.\u00a086 (1991) 68\u2013110.","journal-title":"Adv. Math."},{"key":"9396_CR9","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-7643-8905-5","volume-title":"Lectures on Algebraic Statistics","author":"M Drton","year":"2009","unstructured":"M.\u00a0Drton, B.\u00a0Sturmfels and S.\u00a0Sullivant: Lectures on Algebraic Statistics, Oberwolfach Seminars, 39, Birkh\u00e4user Verlag, Basel, 2009."},{"key":"9396_CR10","doi-asserted-by":"publisher","DOI":"10.1090\/conm\/516\/10173","volume-title":"Smoothness of Gaussian conditional independence models","author":"M Drton","year":"2010","unstructured":"M.\u00a0Drton and H.\u00a0Xiao: Smoothness of Gaussian conditional independence models, Algebraic methods in statistics and probability II, 155\u2013177, Contemporary Mathematics, 516, Amer. Math. Soc., Providence, RI, 2010."},{"key":"9396_CR11","doi-asserted-by":"publisher","first-page":"1152","DOI":"10.1214\/16-AOS1478","volume":"45","author":"S Fallat","year":"2017","unstructured":"S.\u00a0Fallat, S.\u00a0Lauritzen, K.\u00a0Sadeghi, C.\u00a0Uhler, N.\u00a0Wermuth and P.\u00a0Zwiernik: Total positivity in Markov structures, Annals of Statistics 45 (2017) 1152\u20131184.","journal-title":"Annals of Statistics"},{"key":"9396_CR12","doi-asserted-by":"publisher","first-page":"359","DOI":"10.1007\/s00454-012-9470-0","volume":"49","author":"K Fukuda","year":"2013","unstructured":"K.\u00a0Fukuda, H.\u00a0Miyata and S.\u00a0Moriyama: Complete enumeration of small realizable oriented matroids, Discrete Comput. Geom. 49 (2013) 359\u2013381.","journal-title":"Discrete Comput. Geom."},{"key":"9396_CR13","volume-title":"Representation Theory","author":"W Fulton","year":"1991","unstructured":"W.\u00a0Fulton and J.\u00a0Harris: Representation Theory, Graduate Texts in Mathematics 129, Springer, New York, 1991."},{"key":"9396_CR14","unstructured":"P.\u00a0G\u00f6rlach, Y.\u00a0Ren and J.\u00a0Sommars: Detecting tropical defects of polynomial equations, in preparation."},{"key":"9396_CR15","unstructured":"D.\u00a0Grayson and M.\u00a0Stillman: Macaulay2, a software system for research in algebraic geometry, available at \n                    www.math.uiuc.edu\/Macaulay2\/\n                    \n                  ."},{"key":"9396_CR16","doi-asserted-by":"publisher","first-page":"41","DOI":"10.1007\/BF02565846","volume":"57","author":"H Hiller","year":"1982","unstructured":"H.\u00a0Hiller: Combinatorics and intersection of Schubert varieties, Comment. Math. Helv.\u00a057 (1982) 41\u201359.","journal-title":"Comment. Math. Helv."},{"key":"9396_CR17","doi-asserted-by":"publisher","first-page":"634","DOI":"10.1016\/j.jalgebra.2007.01.039","volume":"316","author":"O Holtz","year":"2007","unstructured":"O.\u00a0Holtz and B.\u00a0Sturmfels: Hyperdeterminantal relations among symmetric principal minors, Journal of Algebra 316 (2007) 634\u2013648.","journal-title":"Journal of Algebra"},{"key":"9396_CR18","unstructured":"A.\u00a0Iliev and K.\u00a0Ranestad: Geometry of the Lagrangian Grassmannian\n                           \n                    \n                      \n                    \n                    $$LG(3,6)$$\n                    \n                      \n                        \n                          L\n                          G\n                          (\n                          3\n                          ,\n                          6\n                          )\n                        \n                      \n                    \n                  \n                           with applications to Brill\u2013Noether loci, Michigan Math.\u00a0J. 53(2) (2005) 383\u2013417."},{"key":"9396_CR19","doi-asserted-by":"publisher","first-page":"2177","DOI":"10.1016\/j.jmva.2005.05.010","volume":"97","author":"H Joe","year":"2006","unstructured":"H.\u00a0Joe: Generating random correlation matrices based on partial correlations, J. Multivariate Analysis\u00a097 (2006) 2177\u20132189.","journal-title":"J. Multivariate Analysis"},{"key":"9396_CR20","doi-asserted-by":"publisher","first-page":"419","DOI":"10.1016\/0024-3795(83)90027-7","volume":"52","author":"S Karlin","year":"1983","unstructured":"S.\u00a0Karlin and Y.\u00a0Rinott: M-matrices as covariance matrices of multinormal distributions, Linear Algebra Appl.\u00a052 (1983) 419\u2013438.","journal-title":"Linear Algebra Appl."},{"key":"9396_CR21","unstructured":"K.\u00a0Kaveh and C.\u00a0Manon: Khovanskii bases, higher rank valuations and tropical geometry, \n                    arXiv:1610.00298\n                    \n                  ."},{"key":"9396_CR22","doi-asserted-by":"publisher","first-page":"474010","DOI":"10.1088\/1751-8113\/47\/47\/474010","volume":"47","author":"R Kenyon","year":"2014","unstructured":"R.\u00a0Kenyon and R.\u00a0Pemantle: Principal minors and rhombus tilings, J. Phys. A 47 (2014) 474010, 17 pp.","journal-title":"J. Phys. A"},{"key":"9396_CR23","unstructured":"S.\u00a0Lauritzen, C.\u00a0Uhler and P.\u00a0Zwiernik: Maximum likelihood estimation in Gaussian models under total positivity, Annals of Statistics, to appear."},{"key":"9396_CR24","first-page":"327","volume":"43","author":"R Ln\u011bni\u010dka","year":"2007","unstructured":"R.\u00a0Ln\u011bni\u010dka and F.\u00a0Mat\u00fa\u0161: On Gaussian conditional independence structures, Kybernetika 43 (2007) 327\u2013342.","journal-title":"Kybernetika"},{"key":"9396_CR25","doi-asserted-by":"publisher","DOI":"10.1090\/gsm\/161","volume-title":"Introduction to Tropical Geometry","author":"D Maclagan","year":"2015","unstructured":"D.\u00a0Maclagan and B.\u00a0Sturmfels: Introduction to Tropical Geometry, Graduate Studies in Mathematics 161, American Mathematical Society, Providence, RI, 2015."},{"key":"9396_CR26","unstructured":"S.\u00a0J.\u00a0Maher et\u00a0al.: The SCIP Optimization Suite 4.0, Zuse Institute, Berlin."},{"issue":"4","key":"9396_CR27","doi-asserted-by":"publisher","first-page":"1397","DOI":"10.1093\/qmath\/hat031","volume":"65","author":"D Mayhew","year":"2014","unstructured":"D.\u00a0Mayhew, G.\u00a0Whittle and M.\u00a0Newman: Is the missing axiom of matroid theory lost forever? The Quarterly Journal of Mathematics 65(4) (2014) 1397\u20131415.","journal-title":"The Quarterly Journal of Mathematics"},{"key":"9396_CR28","doi-asserted-by":"crossref","unstructured":"D.\u00a0Mayhew, M.\u00a0Newman and G.\u00a0Whittle: Yes, the \u201cmissing axiom\u201d of matroid theory is lost forever, Transactions of the American Mathematical Society, to appear (2018)","DOI":"10.1090\/tran\/7408"},{"key":"9396_CR29","doi-asserted-by":"publisher","first-page":"64","DOI":"10.1137\/16M107894X","volume":"32","author":"F Mohammadi","year":"2018","unstructured":"F.\u00a0Mohammadi, C.\u00a0Uhler, C.\u00a0Wang and J.\u00a0Yu: Generalized permutohedra from probabilistic graphical models, SIAM Journal on Discrete Mathematics 32 (2018) 64\u201393.","journal-title":"SIAM Journal on Discrete Mathematics"},{"key":"9396_CR30","unstructured":"L.\u00a0Oeding: G-Varieties and the Principal Minors of Symmetric Matrices, PhD Dissertation, Texas A&M University, ProQuest LLC, Ann Arbor, MI, 2009."},{"key":"9396_CR31","doi-asserted-by":"publisher","first-page":"75","DOI":"10.2140\/ant.2011.5.75","volume":"5","author":"L Oeding","year":"2011","unstructured":"L.\u00a0Oeding: Set-theoretic defining equations of the variety of principal minors of symmetric matrices, Algebra and Number Theory 5 (2011) 75\u2013109.","journal-title":"Algebra and Number Theory"},{"key":"9396_CR32","first-page":"1","volume":"18","author":"K Sadeghi","year":"2017","unstructured":"K.\u00a0Sadeghi: Faithfulness of probability distributions and graphs, Journal of Machine Learning Research 18 (2017) 1\u201329.","journal-title":"Journal of Machine Learning Research"},{"key":"9396_CR33","unstructured":"SageMath, the Sage Mathematics Software System (Version\u00a08.0), The Sage Developers, 2017, \n                    http:\/\/www.sagemath.org\n                    \n                  ."},{"key":"9396_CR34","unstructured":"P. \u0160ime\u010dek: Gaussian representation of independence models over four random variables In: Proc.\u00a0COMPSTAT 2006, World Conference on Computational Statistics\u00a017 (A. Rizzi and M. Vichi, eds.), Rome 2006, pp. 1405\u20131412."},{"key":"9396_CR35","doi-asserted-by":"publisher","first-page":"189","DOI":"10.1007\/s10801-005-2513-3","volume":"22","author":"D Speyer","year":"2005","unstructured":"D.\u00a0Speyer and L.\u00a0Williams: The tropical totally positive Grassmannian, J.\u00a0Algebraic Combinatorics 22 (2005) 189\u2013210.","journal-title":"J. Algebraic Combinatorics"},{"key":"9396_CR36","volume-title":"Probabilistic Conditional Independence Structures","author":"M Studeny","year":"2005","unstructured":"M.\u00a0Studeny: Probabilistic Conditional Independence Structures, Information Science and Statistics, Springer, London, 2005."},{"key":"9396_CR37","doi-asserted-by":"publisher","first-page":"351","DOI":"10.1007\/978-0-387-09686-5_10","volume-title":"Emerging Applications of Algebraic Geometry","author":"B Sturmfels","year":"2009","unstructured":"B.\u00a0Sturmfels: Open problems in algebraic statistics, Emerging Applications of Algebraic Geometry. edited by M. Putinar and S.\u00a0Sullivant, Springer, New York (2009), 351\u2013363."},{"key":"9396_CR38","doi-asserted-by":"publisher","first-page":"496","DOI":"10.1016\/j.jcta.2016.06.014","volume":"144","author":"B Sturmfels","year":"2016","unstructured":"B.\u00a0Sturmfels, E.\u00a0Tsukerman and L.\u00a0Williams: Symmetric matrices, Catalan paths, and correlations, J.\u00a0Combinatorial Theory, Ser.\u00a0A 144 (2016) 496\u2013510.","journal-title":"J. Combinatorial Theory, Ser. A"},{"key":"9396_CR39","doi-asserted-by":"publisher","first-page":"1502","DOI":"10.1016\/j.jpaa.2008.11.026","volume":"213","author":"S Sullivant","year":"2009","unstructured":"S.\u00a0Sullivant: Gaussian conditional independence relations have no finite complete characterization, Journal of Pure and Applied Algebra 213 (2009) 1502\u20131506.","journal-title":"Journal of Pure and Applied Algebra"},{"key":"9396_CR40","doi-asserted-by":"crossref","unstructured":"M.\u00a0Thurley: sharpSAT - Counting models with advanced component caching and implicit BCP, Proc. 9th Int. Conf. Theory and Applications of Satisfiability Testing (SAT 2006), (2006) pp. 424\u2013429.","DOI":"10.1007\/11814948_38"},{"issue":"1","key":"9396_CR41","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1145\/2975585","volume":"21","author":"T Toda","year":"2016","unstructured":"T.\u00a0Toda and S.\u00a0Takehide: Implementing efficient all solutions SAT solvers, J. Experimental Algorithmics 21.1 (2016) 1\u201312.","journal-title":"J. Experimental Algorithmics"},{"key":"9396_CR42","doi-asserted-by":"publisher","first-page":"403","DOI":"10.1112\/jlms\/s2-18.3.403","volume":"18","author":"P V\u00e1mos","year":"1978","unstructured":"P.\u00a0V\u00e1mos: The missing axiom of matroid theory is lost forever, Journal of the London Mathematical Society 18 (1978) 403\u2013408.","journal-title":"Journal of the London Mathematical Society"}],"container-title":["Foundations of Computational Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10208-018-9396-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10208-018-9396-x\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10208-018-9396-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,9,4]],"date-time":"2019-09-04T23:05:40Z","timestamp":1567638340000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10208-018-9396-x"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,9,5]]},"references-count":42,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2019,8]]}},"alternative-id":["9396"],"URL":"https:\/\/doi.org\/10.1007\/s10208-018-9396-x","relation":{},"ISSN":["1615-3375","1615-3383"],"issn-type":[{"value":"1615-3375","type":"print"},{"value":"1615-3383","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,9,5]]},"assertion":[{"value":"20 October 2017","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"28 May 2018","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"31 May 2018","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 September 2018","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}