CONTEMPORARY MATHEMATICS

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1 CONTEMPORARY MATHEMATICS 452 Integer Points in Polyhedra- Geometry, Number Theory, Representation Theory, Algebra, Optimization, Statistics AMS-IMS-SIAM Joint Summer Research Conference June 11-15, 2006 Snowbird, Utah Matthias Beck Christian Haase Bruce Reznick Michele Vergne Volkmar Welker Ruriko Yoshida Editors

2 Integer Points in Polyhedra - Geometry, Number Theory, Representation Theory, Algebra, Optimization, Statistics

3 CoNTEMPORARY MATHEMATICS 452 Integer Points in Polyhedra - Geometry, Number Theory, Representation Theory, Algebra, Optimization, Statistics AMS-IMS-SIAM Joint Summer Research Conference June 11-15, 2006 Snowbird, Utah Matthias Beck Christian Haase Bruce Reznick Michele Vergne Volkmar Welker Ruriko Yoshida Editors American Mathematical Society Providence, Rhode Island

4 Editorial Board Dennis DeThrck, managing editor George Andrews Andreas Blass Abel Klein This volume contains the proceedings of an AMS-IMS-SIAM Joint Summer Research Conference on Integer Points in Polyhedra-Geometry, Number Theory, Representation Theory, Algebra, Optimization, Statistics, held in Snowbird, Utah, from June 11-15, 2006, with support from the National Science Foundation, grant DMS Mathematics Subject Classification. Primary 52B20; Secondary 05A15, 11D04, 11H06, 11P21, 13P10, 14M25, 14J32, 52C07, 90C10. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. Library of Congress Cataloging-in-Publication Data AMS-IMS-SIAM Joint Summer Research Conference on Integer Points in Polyhedra-Geometry, Number Theory, Representation Theory, Algebra, Optimization, Statistics (2006 : Snowbird, Utah) Integer points in polyhedra : AMS-IMS-SIAM Joint Summer Research Conference on Integer Points in Polyhedra-Geometry, Number Theory, Representation Theory, Algebra, Optimization, Statistics, June 11-15, 2006, Snowbird, Utah/ Matthias Beck... [eta!.], editors. p. em. -(Contemporary mathematics, ISSN ; v. 452) Includes bibliographical references. ISBN (alk. paper) 1. Ordered algebraic structures-congresses. 2. Semigroups-Congresses. 3. Lattice theory- Congresses. 4. Diophantine equations-congresses. I. Beck, Matthias. II. Title. QA174.A '3--dc Copying and reprinting. Material in this book may be reproduced by any means for educational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledgment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island , USA. Requests can also be made by to reprint-permissionc!iams. org. Excluded from these provisions is material in articles for which the author holds copyright. In such cases, requests for permission to use or reprint should be addressed directly to the author(s). (Copyright ownership is indicated in the notice in the lower right-hand corner of the first page of each article.) 2008 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Copyright of individual articles may revert to the public domain 28 years after publication. Contact the AMS for copyright status of individual articles. Printed in the United States of America. The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http: I /www. ams. org/

5 Contents Preface List of Participants Lattice reformulation of integer programming problems KAREN AARDAL 1 Local Euler-Maclaurin expansion of Barvinok valuations and Ehrhart coefficients of a rational polytope VELLEDA BALDONI, NICOLE BERLINE, and MICHELE VERGNE 15 Combinatorial aspects of mirror symmetry VICTOR BATYREV and BENJAMIN NILL 35 Ehrhart polynomial roots and Stanley's non-negativity theorem BENJAMIN BRAUN and MIKE DEVELIN 67 Enumeration of integer solutions to linear inequalities defined by digraphs J. WILLIAM DAVIS, ERWIN D'SOUZA, SUNYOUNG LEE, and CARLA D. SAVAGE 79 Perfect Delaunay polytopes and perfect quadratic functions on lattices ROBERT ERDAHL, ANDREI 0RDINE, and KONSTANTIN RYBNIKOV 93 Quasi-period collapse and GLn(Z)-scissors congruence in rational polytopes CHRISTIAN HAASE and TYRRELL B. McALLISTER 115 Quadratic Grobner bases arising from combinatorics TAKAYUKI HIBI and HIDEFUMI 0HSUGI 123 Ehrhart analogue of the h-vector KALLE KARU 139 Saturation points on faces of a rational polyhedral cone AKIMICHI TAKEMURA AND RURIKO YOSHIDA 147 An explicit formulation for two dimensional vector partition functions ZHIQIANG Xu 163 Let me tell you my favorite lattice-point problem... MATTHIAS BECK and BRUCE REZNICK (EDITORS) 179 vii ix v

6 Preface This volume contains original research and survey papers stemming from the AMS- IMS-SIAM Joint Summer Research Conference "Integer points in Polyhedra- Geometry, Number Theory, Representation Theory, Algebra, Optimization, Statistics.", which took place from June 11 to June 15, 2006, in Snowbird, Utah. The conference website http: I /math. sf su. edu/beck/ src06. html contains the program and pictures of the conference. The volume is a cross section of recent advances connected to lattice-point questions. Similar to the talks given at the conference, topics range from commutative algebra to optimization, from discrete geometry to statistics, from mirror symmetry to geometry of numbers. We thank the AMS, IMS, and SIAM for their generous support, which allowed us, among other things, to sponsor undergraduate and graduate students for the conference. We would like to express our deep gratitude to Wayne Drady, the local AMS organizer in Snowbird, who made sure--once more in his most pleasant way-that our conference ran smoothly in every possible aspect. We are grateful to all contributors and the anonymous referees for invaluable help. Many thanks to Christine Thivierge and her staff for their patience and for their assistance in preparing these proceedings. Finally, thanks to all participants of the conference, for making it such a pleasant and productive event. Matthias Beck Christian Haase Bruce Reznick Michele Vergne Volkmar Welker Ruriko Yoshida vii

7 List of Participants Karen Aardal Centrum voor Wiskunde en Informatica Amsterdam, Netherlands Karen.Aardal~cwi.nl Jose Agapito Instituto Superior Tecnico Lisboa, Portugal agapito~math.ist.utl.pt Alexander Barvinok University of Michigan Ann Arbor, Michigan, USA barvinok~ath.lsa.umich.edu Victor Batyrev Universitat Tiibingen Tiibingen, Germany victor.batyrev~uni-tuebingen.de Matthias Beck San Francisco State University San Francisco, CA, USA beck~ath.sfsu.edu Nicole Berline Ecole Polytechnique Palaiseau, France Nicole.Berline~ath. polytechnique.fr Sara Billey University of Washington Seattle, WA, USA billey~ath.washington.edu Tristram Bogart University of Washington Seattle, WA, USA bogart~ath.washington.edu Ben Braun Washington University St. Louis, MO, USA bbraun~wustl.edu Zhao Chen New York City College of Technology Brooklyn, NY, USA zchen~citytech.cuny.edu Mike Develin Berkeley, CA, USA develin~ath.berkeley.edu Persi Diaconis Stanford University Palo Alto, CA, USA diaconis~math.stanford.edu Ricardo Diaz University of Northern Colorado Greeley, CO, USA ricardo.diaz~unco.edu Han Doung University of Illinois Urbana, IL, USA handuong~ath.uiuc.edu Christian Haase Freie Universitat Berlin Berlin, Germany christian.haase~ath.fu-berlin.de Takayuki Hibi Osaka University Toyonaka, Osaka, Japan hibi~ath.sci.osaka-u.ac.jp ix

8 X PARTICIPANTS Serkan Ho ten San Francisco State University San Francisco, CA, USA Peter Huggins Berkeley, CA, USA phuggins~math.berkeley.edu Kalle Karu University of British Columbia Vancouver, B.C., Canada karu~math.ubc.ca Alain Lascoux IGM, Universite de Marne-la-Vallee Marne-la-Vallee, France al~univ-mlv.fr Jesus De Loera Davis, CA, USA Diane Maclagan Rutgers University Picataway, N J, USA maclagan~math.rutgers.edu Evgeny Materov University of Massachusetts Amherst, MA, USA materov@math.umass.edu Tyrrell McAllister Davis, CA, USA tmcal@math.ucdavis.edu Ilarion Melnikov Duke University Durham, NC, USA lmel~cgtp.duke.edu Ezra Miller University of Minnesota Minneapolis, MN, USA ezra@math.umn.edu Satoshi Murai Osaka University Toyonaka, Osaka, Japan iav042ms~ecs.cmc.osaka-u.ac.jp Benjamin Nill Freie Universitiit Berlin Berlin, Germany nill~math.fu-berlin.de Andreas Paffenholz Freie Universitiit Berlin Berlin, Germany paffenho@math.fu-berlin.de Gabor Pataki University of North Carolina Chapel Hill, NC, USA gabor@unc.edu Sam Payne University of Michigan Ann Arbor, MI, USA sdpayne@umich.edu Bruce Reznick University of Illinois Urbana, IL, USA reznick~math.uiuc.edu Carla Savage NC State University Raleigh, NC, USA savage@csc.ncsu.edu I van Soprunov University of Massachusetts Amherst, MA, USA isoprou@math.umass.edu Akimuchi Takemura University of Tokyo Bunkyo, Tokyo, Japan takemura@stat.t.u-tokyo.ac.jp Michael von Thaden University of Osnabriick Osnabriick, Germany Michael-v-T@t-online.de

9 PARTICIPANTS xi Ellen Veomett University of Michigan Ann Arbor, MI, USA eveomettqumich.edu Sven Verdoolaege Universiteit Leiden Lei den, Netherlands Michele Vergne Ecole Polytechnique Palaiseau, France vergneqmath.polytechnique.fr Jonathan Weitsman Santa Cruz, CA, USA weitsmanqcats.ucsc.edu Kevin Woods Berkeley, CA, USA kwoodsqmath.berkeley.edu Zhiquiang Xu Chinese Academy of Sciences Beijing, China xuzq~lsec.cc.ac.cn Ruriko Yoshida Duke University Durham, NC, USA rurikoqmath.duke.edu Josephine Yu Berkeley, CA, USA jyu~math.berkeley.edu

10 The AMS-IMS-SIAM Joint Summer Research Conference "Integer Points in Polyhedra- Geometry, Number Theory, Representation Theory, Algebra, Optimization, Statistics" was held in Snowbird, Utah in June This proceedings volume contains research and survey articles originating from the conference. The volume is a cross section of recent advances connected to lattice-point questions. Similar to the talks given at the conference, topics range from commutative algebra to optimization, from discrete geometry to statistics, from mirror symmetry to geometry of numbers. The book is suitable for researchers and graduate students interested in combinatorial aspects of the above fields. ISBN