{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,8]],"date-time":"2025-10-08T15:39:16Z","timestamp":1759937956549,"version":"3.37.3"},"reference-count":40,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2020,8,12]],"date-time":"2020-08-12T00:00:00Z","timestamp":1597190400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2020,8,12]],"date-time":"2020-08-12T00:00:00Z","timestamp":1597190400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100001691","name":"Japan Society for the Promotion of Science","doi-asserted-by":"publisher","award":["18H01134","19K14505"],"award-info":[{"award-number":["18H01134","19K14505"]}],"id":[{"id":"10.13039\/501100001691","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001691","name":"Japan Society for the Promotion of Science","doi-asserted-by":"publisher","award":["19J00312"],"award-info":[{"award-number":["19J00312"]}],"id":[{"id":"10.13039\/501100001691","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Discrete Comput Geom"],"published-print":{"date-parts":[[2021,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>A lattice polytope<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathscr {P} \\subset \\mathbb {R}^d$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mrow><mml:mi>P<\/mml:mi><mml:mo>\u2282<\/mml:mo><mml:msup><mml:mrow><mml:mi>R<\/mml:mi><\/mml:mrow><mml:mi>d<\/mml:mi><\/mml:msup><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>is called a locally anti-blocking polytope if for any closed orthant<jats:inline-formula><jats:alternatives><jats:tex-math>$${\\mathbb R}^d_{\\varepsilon }$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:msubsup><mml:mrow><mml:mi>R<\/mml:mi><\/mml:mrow><mml:mi>\u03b5<\/mml:mi><mml:mi>d<\/mml:mi><\/mml:msubsup><\/mml:math><\/jats:alternatives><\/jats:inline-formula>in<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathbb {R}^d$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:msup><mml:mrow><mml:mi>R<\/mml:mi><\/mml:mrow><mml:mi>d<\/mml:mi><\/mml:msup><\/mml:math><\/jats:alternatives><\/jats:inline-formula>,<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathscr {P} \\cap \\mathbb {R}^d_{\\varepsilon }$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mrow><mml:mi>P<\/mml:mi><mml:mo>\u2229<\/mml:mo><mml:msubsup><mml:mrow><mml:mi>R<\/mml:mi><\/mml:mrow><mml:mi>\u03b5<\/mml:mi><mml:mi>d<\/mml:mi><\/mml:msubsup><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>is unimodularly equivalent to an anti-blocking polytope by reflections of coordinate hyperplanes. We give a formula for the<jats:inline-formula><jats:alternatives><jats:tex-math>$$h^*$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:msup><mml:mi>h<\/mml:mi><mml:mo>\u2217<\/mml:mo><\/mml:msup><\/mml:math><\/jats:alternatives><\/jats:inline-formula>-polynomials of locally anti-blocking lattice polytopes. In particular, we discuss the<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\gamma $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>\u03b3<\/mml:mi><\/mml:math><\/jats:alternatives><\/jats:inline-formula>-positivity of<jats:inline-formula><jats:alternatives><jats:tex-math>$$h^*$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:msup><mml:mi>h<\/mml:mi><mml:mo>\u2217<\/mml:mo><\/mml:msup><\/mml:math><\/jats:alternatives><\/jats:inline-formula>-polynomials of locally anti-blocking reflexive polytopes.<\/jats:p>","DOI":"10.1007\/s00454-020-00236-6","type":"journal-article","created":{"date-parts":[[2020,8,12]],"date-time":"2020-08-12T17:03:59Z","timestamp":1597251839000},"page":"701-722","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["The $$h^*$$-Polynomials of Locally Anti-Blocking Lattice Polytopes and Their $$\\gamma $$-Positivity"],"prefix":"10.1007","volume":"66","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6403-1119","authenticated-orcid":false,"given":"Hidefumi","family":"Ohsugi","sequence":"first","affiliation":[]},{"given":"Akiyoshi","family":"Tsuchiya","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2020,8,12]]},"reference":[{"issue":"1","key":"236_CR1","doi-asserted-by":"publisher","first-page":"360","DOI":"10.1137\/090749293","volume":"25","author":"F Ardila","year":"2011","unstructured":"Ardila, F., Beck, M., Ho\u015ften, S., Pfeifle, J., Seashore, K.: Root polytopes and growth series of root lattices. SIAM J. Discrete Math. 25(1), 360\u2013378 (2011)","journal-title":"SIAM J. Discrete Math."},{"key":"236_CR2","unstructured":"Athanasiadis, Ch.A.: Gamma-positivity in combinatorics and geometry. S\u00e9minaire Lotharingien de Combinatoire 77, #\u00a0B77i (2016\u20132018)"},{"issue":"3","key":"236_CR3","first-page":"493","volume":"3","author":"VV Batyrev","year":"1994","unstructured":"Batyrev, V.V.: Dual polyhedra and mirror symmetry for Calabi\u2013Yau hypersurfaces in toric varieties. J. Algebr. Geom. 3(3), 493\u2013535 (1994)","journal-title":"J. Algebr. Geom."},{"issue":"1","key":"236_CR4","doi-asserted-by":"publisher","first-page":"#\u00a015","DOI":"10.37236\/1153","volume":"13","author":"B Braun","year":"2006","unstructured":"Braun, B.: An Ehrhart series formula for reflexive polytopes. Electron. J. Comb. 13(1), #\u00a015 (2006)","journal-title":"Electron. J. Comb."},{"issue":"1","key":"236_CR5","doi-asserted-by":"publisher","first-page":"65","DOI":"10.1016\/j.jcta.2006.03.003","volume":"114","author":"W Bruns","year":"2007","unstructured":"Bruns, W., R\u00f6mer, T.: $$h$$-Vectors of Gorenstein polytopes. J. Comb. Theory Ser. A 114(1), 65\u201376 (2007)","journal-title":"J. Comb. Theory Ser. A"},{"issue":"1966","key":"236_CR6","doi-asserted-by":"publisher","first-page":"2369","DOI":"10.1098\/rspa.1997.0126","volume":"453","author":"JH Conway","year":"1997","unstructured":"Conway, J.H., Sloane, N.J.A.: Low-dimensional lattices. VII. Coordination sequences. Proc. R. Soc. Lond. Ser. A 453(1966), 2369\u20132389 (1997)","journal-title":"Proc. R. Soc. Lond. Ser. A"},{"key":"236_CR7","volume-title":"Toric Varieties. Graduate Studies in Mathematics","author":"DA Cox","year":"2011","unstructured":"Cox, D.A., Little, J.B., Schenck, H.K.: Toric Varieties. Graduate Studies in Mathematics, vol. 124. American Mathematical Society, Providence (2011)"},{"key":"236_CR8","unstructured":"D\u2019Al\u00ec, A., Delucchi, E., Micha\u0142ek, M.: Many faces of symmetric edge polytopes (2019). arXiv:1910.05193"},{"key":"236_CR9","volume-title":"Graph Theory. Graduate Texts in Mathematics","author":"R Diestel","year":"2010","unstructured":"Diestel, R.: Graph Theory. Graduate Texts in Mathematics, vol. 173. Springer, Heidelberg (2010)"},{"key":"236_CR10","unstructured":"Ehrhart, E.: Polynom\u00eas Arithm\u00e9tiques et M\u00e9thode des Poly\u00e8dres en Combinatorie International Series of Numerical Mathematics, vol. 35. Birkh\u00e4user, Basel\u2013Stuttgart (1977)"},{"key":"236_CR11","doi-asserted-by":"publisher","first-page":"168","DOI":"10.1007\/BF01584085","volume":"1","author":"DR Fulkerson","year":"1971","unstructured":"Fulkerson, D.R.: Blocking and anti-blocking pairs of polyhedra. Math. Program. 1, 168\u2013194 (1971)","journal-title":"Math. Program."},{"key":"236_CR12","doi-asserted-by":"publisher","first-page":"50","DOI":"10.1016\/0095-8956(72)90032-9","volume":"12","author":"DR Fulkerson","year":"1972","unstructured":"Fulkerson, D.R.: Anti-blocking polyhedra. J. Comb. Theory Ser. B 12, 50\u201371 (1972)","journal-title":"J. Comb. Theory Ser. B"},{"issue":"2","key":"236_CR13","doi-asserted-by":"publisher","first-page":"269","DOI":"10.1007\/s00454-005-1171-5","volume":"34","author":"\u015aR Gal","year":"2005","unstructured":"Gal, \u015a.R.: Real Root Conjecture fails for five- and higher-dimensional spheres. Discrete Comput. Geom. 34(2), 269\u2013284 (2005)","journal-title":"Discrete Comput. Geom."},{"key":"236_CR14","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-95349-6","volume-title":"Binomial Ideals. Graduate Texts in Mathematics","author":"J Herzog","year":"2018","unstructured":"Herzog, J., Hibi, T., Ohsugi, H.: Binomial Ideals. Graduate Texts in Mathematics, vol. 279. Springer, Cham (2018)"},{"issue":"2","key":"236_CR15","doi-asserted-by":"publisher","first-page":"237","DOI":"10.1007\/BF01204726","volume":"12","author":"T Hibi","year":"1992","unstructured":"Hibi, T.: Dual polytopes of rational convex polytopes. Combinatorica 12(2), 237\u2013240 (1992)","journal-title":"Combinatorica"},{"key":"236_CR16","unstructured":"Hibi, T., Matsuda, K., Tsuchiya, A.: Gorenstein Fano polytopes arising from order polytopes and chain polytopes. arXiv:1507.03221"},{"key":"236_CR17","doi-asserted-by":"publisher","DOI":"10.1307\/mmj\/1585792887","author":"T Hibi","year":"2020","unstructured":"Hibi, T., Ohsugi, H., Tsuchiya, A.: Integer decomposition property for Cayley sums of order and stable set polytopes. Michigan Math. J. (2020). https:\/\/doi.org\/10.1307\/mmj\/1585792887","journal-title":"Michigan Math. J."},{"issue":"16","key":"236_CR18","doi-asserted-by":"publisher","first-page":"2619","DOI":"10.1002\/mana.201600396","volume":"290","author":"T Hibi","year":"2017","unstructured":"Hibi, T., Tsuchiya, A.: Facets and volume of Gorenstein Fano polytopes. Math. Nachr. 290(16), 2619\u20132628 (2017)","journal-title":"Math. Nachr."},{"key":"236_CR19","doi-asserted-by":"publisher","first-page":"233","DOI":"10.1016\/j.jcta.2018.02.012","volume":"157","author":"T Hibi","year":"2018","unstructured":"Hibi, T., Tsuchiya, A.: Reflexive polytopes arising from perfect graphs. J. Comb. Theory Ser. A 157, 233\u2013246 (2018)","journal-title":"J. Comb. Theory Ser. A"},{"issue":"3","key":"236_CR20","doi-asserted-by":"publisher","first-page":"579","DOI":"10.1215\/21562261-3089073","volume":"55","author":"A Higashitani","year":"2015","unstructured":"Higashitani, A.: Smooth Fano polytopes arising from finite directed graphs. Kyoto J. Math. 55(3), 579\u2013592 (2015)","journal-title":"Kyoto J. Math."},{"issue":"3","key":"236_CR21","doi-asserted-by":"publisher","first-page":"763","DOI":"10.1112\/S0025579319000147","volume":"65","author":"A Higashitani","year":"2019","unstructured":"Higashitani, A., Jochemko, K., Micha\u0142ek, M.: Arithmetic aspects of symmetric edge polytopes. Mathematika 65(3), 763\u2013784 (2019)","journal-title":"Mathematika"},{"key":"236_CR22","doi-asserted-by":"publisher","first-page":"823","DOI":"10.1016\/j.aim.2013.06.001","volume":"244","author":"T K\u00e1lm\u00e1n","year":"2013","unstructured":"K\u00e1lm\u00e1n, T.: A version of Tutte\u2019s polynomial for hypergraphs. Adv. Math. 244, 823\u2013873 (2013)","journal-title":"Adv. Math."},{"key":"236_CR23","doi-asserted-by":"publisher","DOI":"10.1007\/s00454-020-00199-8","author":"F Kohl","year":"2020","unstructured":"Kohl, F., Olsen, M., Sanyal, R.: Unconditional reflexive polytopes. Discrete Comput. Geom. (2020). https:\/\/doi.org\/10.1007\/s00454-020-00199-8","journal-title":"Discrete Comput. Geom."},{"issue":"6","key":"236_CR24","doi-asserted-by":"publisher","first-page":"1209","DOI":"10.4310\/ATMP.2000.v4.n6.a2","volume":"4","author":"M Kreuzer","year":"2000","unstructured":"Kreuzer, M., Skarke, H.: Complete classification of reflexive polyhedra in four dimensions. Adv. Theor. Math. Phys. 4(6), 1209\u20131230 (2000)","journal-title":"Adv. Theor. Math. Phys."},{"issue":"5","key":"236_CR25","doi-asserted-by":"publisher","first-page":"1022","DOI":"10.4153\/CJM-1991-058-4","volume":"43","author":"JC Lagarias","year":"1991","unstructured":"Lagarias, J.C., Ziegler, G.M.: Bounds for lattice polytopes containing a fixed number of interior points in a sublattice. Can. J. Math. 43(5), 1022\u20131035 (1991)","journal-title":"Can. J. Math."},{"issue":"4","key":"236_CR26","doi-asserted-by":"publisher","first-page":"721","DOI":"10.1007\/s10801-011-0290-8","volume":"34","author":"T Matsui","year":"2011","unstructured":"Matsui, T., Higashitani, A., Nagazawa, Y., Ohsugi, H., Hibi, T.: Roots of Ehrhart polynomials arising from graphs. J. Algebr. Comb. 34(4), 721\u2013749 (2011)","journal-title":"J. Algebr. Comb."},{"key":"236_CR27","doi-asserted-by":"crossref","unstructured":"Nill, B.: Classification of pseudo-symmetric simplicial reflexive polytopes. In: Algebraic and Geometric Combinatorics (Anogia 2005). Contemp. Math., vol. 423, pp. 269\u2013282. American Mathematical Society, Providence (2006)","DOI":"10.1090\/conm\/423\/08082"},{"key":"236_CR28","doi-asserted-by":"publisher","first-page":"153","DOI":"10.1215\/00277630-2857555","volume":"216","author":"H Ohsugi","year":"2014","unstructured":"Ohsugi, H., Hibi, T.: Centrally symmetric configurations of integer matrices. Nagoya Math. J. 216, 153\u2013170 (2014)","journal-title":"Nagoya Math. J."},{"issue":"3","key":"236_CR29","doi-asserted-by":"publisher","first-page":"624","DOI":"10.1007\/s00454-012-9395-7","volume":"47","author":"H Ohsugi","year":"2012","unstructured":"Ohsugi, H., Shibata, K.: Smooth Fano polytopes whose Ehrhart polynomial has a root with large real part. Discrete Comput. Geom. 47(3), 624\u2013628 (2012)","journal-title":"Discrete Comput. Geom."},{"key":"236_CR30","doi-asserted-by":"publisher","first-page":"485","DOI":"10.1007\/s11856-020-2012-1","volume":"237","author":"H Ohsugi","year":"2020","unstructured":"Ohsugi, H., Tsuchiya, A.: Enriched chain polytopes. Israel J. Math. 237, 485\u2013500 (2020)","journal-title":"Israel J. Math."},{"key":"236_CR31","unstructured":"Ohsugi, H., Tsuchiya, A.: Reflexive polytopes arising from bipartite graphs with $$\\gamma $$-positivity associated to interior polynomials. Selecta Math. (N.S.), to appear"},{"issue":"12","key":"236_CR32","doi-asserted-by":"publisher","first-page":"905","DOI":"10.1524\/zkri.1995.210.12.905","volume":"210","author":"M O\u2019Keeffe","year":"1995","unstructured":"O\u2019Keeffe, M.: Coordination sequences for lattices. Zeitschrift f\u00fcr Kristallographie 210(12), 905\u2013908 (1995)","journal-title":"Zeitschrift f\u00fcr Kristallographie"},{"issue":"2","key":"236_CR33","doi-asserted-by":"publisher","first-page":"561","DOI":"10.1016\/j.aim.2006.05.016","volume":"209","author":"TK Petersen","year":"2007","unstructured":"Petersen, T.K.: Enriched $$P$$-partitions and peak algebras. Adv. Math. 209(2), 561\u2013610 (2007)","journal-title":"Adv. Math."},{"key":"236_CR34","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4939-3091-3","volume-title":"Eulerian Numbers. Birkh\u00e4user Advanced Texts","author":"TK Petersen","year":"2015","unstructured":"Petersen, T.K.: Eulerian Numbers. Birkh\u00e4user Advanced Texts. Basler Lehrb\u00fccher. Birkh\u00e4user, New York (2015)"},{"key":"236_CR35","volume-title":"Theory of Linear and Integer Programming. Wiley-Interscience Series in Discrete Mathematics","author":"A Schrijver","year":"1986","unstructured":"Schrijver, A.: Theory of Linear and Integer Programming. Wiley-Interscience Series in Discrete Mathematics. Wiley, Chichester (1986)"},{"key":"236_CR36","doi-asserted-by":"publisher","first-page":"333","DOI":"10.1016\/S0167-5060(08)70717-9","volume":"6","author":"RP Stanley","year":"1980","unstructured":"Stanley, R.P.: Decompositions of rational convex polytopes. Ann. Discrete Math. 6, 333\u2013342 (1980)","journal-title":"Ann. Discrete Math."},{"issue":"1","key":"236_CR37","doi-asserted-by":"publisher","first-page":"9","DOI":"10.1007\/BF02187680","volume":"1","author":"RP Stanley","year":"1986","unstructured":"Stanley, R.P.: Two poset polytopes. Discrete Comput. Geom. 1(1), 9\u201323 (1986)","journal-title":"Discrete Comput. Geom."},{"issue":"2","key":"236_CR38","doi-asserted-by":"publisher","first-page":"763","DOI":"10.1090\/S0002-9947-97-01804-7","volume":"349","author":"JR Stembridge","year":"1997","unstructured":"Stembridge, J.R.: Enriched $$P$$-partitions. Trans. Am. Math. Soc. 349(2), 763\u2013788 (1997)","journal-title":"Trans. Am. Math. Soc."},{"issue":"4","key":"236_CR39","doi-asserted-by":"publisher","first-page":"875","DOI":"10.1007\/s00026-018-0405-1","volume":"22","author":"A Tsuchiya","year":"2018","unstructured":"Tsuchiya, A.: Volume, facets and dual polytopes of twinned chain polytopes. Ann. Comb. 22(4), 875\u2013884 (2018)","journal-title":"Ann. Comb."},{"key":"236_CR40","unstructured":"Wang, C., Yu, J.: Toric $$h$$-vectors and Chow Betti numbers of dual hypersimplices (2017). arXiv:1707.04581"}],"container-title":["Discrete &amp; Computational Geometry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00454-020-00236-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00454-020-00236-6\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00454-020-00236-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,11,6]],"date-time":"2022-11-06T18:28:58Z","timestamp":1667759338000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00454-020-00236-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,8,12]]},"references-count":40,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2021,9]]}},"alternative-id":["236"],"URL":"https:\/\/doi.org\/10.1007\/s00454-020-00236-6","relation":{},"ISSN":["0179-5376","1432-0444"],"issn-type":[{"type":"print","value":"0179-5376"},{"type":"electronic","value":"1432-0444"}],"subject":[],"published":{"date-parts":[[2020,8,12]]},"assertion":[{"value":"16 June 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"25 May 2020","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 July 2020","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"12 August 2020","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}