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@article{greenwade93,
	Author = {George D. Greenwade},
	Date-Modified = {2016-08-11 21:04:25 +0000},
	Journal = {TUGBoat},
	Number = {3},
	Pages = {342--351},
	Title = {The {C}omprehensive {T}ex {A}rchive {N}etwork ({CTAN})},
	Url = {https://www.google.com},
	Volume = {14},
	Year = {1993},
	Bdsk-Url-1 = {https://www.google.com}}

@article{MR4,
AUTHOR = {Corliss, J. J.},
     TITLE = {Upper limits to the real roots of a real algebraic equation},
   JOURNAL = {Amer. Math. Monthly},
  FJOURNAL = {The American Mathematical Monthly},
    VOLUME = {46},
      YEAR = {1939},
     PAGES = {334--338},
      ISSN = {0002-9890},
   MRCLASS = {09.0X},
  MRNUMBER = {0000004},
MRREVIEWER = {O. Sz\'asz},
       DOI = {10.2307/2302887}}

@misc{MP,
    AUTHOR = {Madhava, I. and Pingala, A.},
    EPRINT = {0201.00000},
    EPRINTTYPE = {arxiv},
    TITLE = {Title of the arXiv preprint},
    YEAR = {0002}}


@article {ehrhartpolynomial,
    AUTHOR = {Ehrhart, E.},
     TITLE = {Sur les poly\`edres rationnels homoth\'etiques \`a {$n$}\
              dimensions},
   JOURNAL = {C. R. Acad. Sci. Paris},
    VOLUME = {254},
      YEAR = {1962},
     PAGES = {616--618},
   MRCLASS = {10.25 (52.10)},
  MRNUMBER = {24 \#A714},
}

@article {stapledonfreesums,
    AUTHOR = {Stapledon, A.},
     TITLE = {Counting lattice points in free sums of polytopes},
   JOURNAL = {J. Combin. Theory Ser. A},
  FJOURNAL = {Journal of Combinatorial Theory. Series A},
    VOLUME = {151},
      YEAR = {2017},
     PAGES = {51--60},
      ISSN = {0097-3165},
   MRCLASS = {52B20},
 MRNUMBER = {3663488},
MRREVIEWER = {Matthias Beck},
}

@misc{fernandespinaalfonsinrobins,
      title={On the period collapse of a family of {E}hrhart quasi-polynomials}, 
      author={C. G. Fernandes and J. C. de Pina and J. L. Ram\'irez Alfons\'in and S. Robins},
      year={2021},
      EPRINTTYPE = {arxiv},
      EPRINT = {2104.11025},
}

@article {baldoniberlinekoeppevergnerealdilation,
    AUTHOR = {V. Baldoni and N. Berline and M. K{\"o}ppe and M. Vergne},
     TITLE = {Intermediate sums on polyhedra: computation and real {E}hrhart theory},
   JOURNAL = {Mathematika},
  FJOURNAL = {Mathematika. A Journal of Pure and Applied Mathematics},
    VOLUME = {59},
      YEAR = {2013},
    NUMBER = {1},
     PAGES = {1--22},
      ISSN = {0025-5793},
   MRCLASS = {52C07 (05A15 52B45 68R05 68U05 90C10)},
 MRNUMBER = {3028168},
MRREVIEWER = {Alexander I. Barvinok},
}

@article {mcallisterwoods,
    AUTHOR = {McAllister, T. B. and Woods, K. M.},
     TITLE = {The minimum period of the {E}hrhart quasi-polynomial of a rational polytope},
   JOURNAL = {J. Combin. Theory Ser. A},
  FJOURNAL = {Journal of Combinatorial Theory. Series A},
    VOLUME = {109},
      YEAR = {2005},
    NUMBER = {2},
     PAGES = {345--352},
note = {{\tt arXiv:math.CO/0310255}},
      ISSN = {0097-3165},
     CODEN = {JCBTA7},
   MRCLASS = {52B05 (05E10)},
  MRNUMBER = {MR2121031},
}


@misc{cristofarogardiner,
    AUTHOR = {D. Cristofaro-Gardiner and T. Xueshan Li and R. P. Stanley},
    EPRINT = {1509.01887},
    EPRINTTYPE = {arxiv},
    TITLE = {New examples of period collapse},
    YEAR = {2015}}

@article{rationalbetkemcmullen,
      title={Decompositions of {E}hrhart $h^*$-polynomials for rational polytopes}, 
      author={Beck, M. and Braun, B. and Vindas-Mel\'endez, A. R.},
      year={2022},
      journal={Discrete Comput. Geom},
      number = {68},
      pages = {50--71},
}


@article {betkemcmullen,
    AUTHOR = {Betke, U. and McMullen, P.},
     TITLE = {Lattice points in lattice polytopes},
   JOURNAL = {Monatsh. Math.},
  FJOURNAL = {Monatshefte f\"ur Mathematik},
    VOLUME = {99},
      YEAR = {1985},
    NUMBER = {4},
     PAGES = {253--265},
      ISSN = {0026-9255},
     CODEN = {MNMTA2},
   MRCLASS = {52A43 (11H06 52A25)},
  MRNUMBER = {MR799674 (87e:52019)},
MRREVIEWER = {J. M. Wills},
}

@incollection {BatyrevNill,
    AUTHOR = {Batyrev, V. V. and Nill, B.},
     TITLE = {Combinatorial aspects of mirror symmetry},
 BOOKTITLE = {Integer Points in Polyhedra --- Geometry, Number Theory,
              Representation Theory, Algebra, Optimization, Statistics},
    SERIES = {Contemp. Math.},
    VOLUME = {452},
     PAGES = {35--66},
 PUBLISHER = {Amer. Math. Soc., Providence, RI},
      YEAR = {2008},
   MRCLASS = {14J32 (14M25 52B20)},
  MRNUMBER = {2405763},
MRREVIEWER = {Julianna Tymoczko},
}

@book {brunsgubeladzektheory,
    AUTHOR = {Bruns, W. and Gubeladze, J.},
     TITLE = {Polytopes, rings, and {$K$}-theory},
    SERIES = {Springer Monographs in Mathematics},
 PUBLISHER = {Springer},
   ADDRESS = {Dordrecht},
      YEAR = {2009},
     PAGES = {xiv+461},
      ISBN = {978-0-387-76355-2},
   MRCLASS = {19-02 (11H06 13F45 14M25 52-01 52B20)},
  MRNUMBER = {2508056 (2010d:19001)},
MRREVIEWER = {T. Oda},
}

@book {ziegler,
    AUTHOR = {Ziegler, G. M.},
     TITLE = {Lectures on {P}olytopes},
 PUBLISHER = {Springer-Verlag},
   ADDRESS = {New York},
      YEAR = {1995},
     PAGES = {x+370},
      ISBN = {0-387-94365-X},
 note =	 {Revised edition, 1998; ``Updates, corrections, and more" at {\tt www.math.tu-berlin.de/$\sim$ziegler}},
   MRCLASS = {52Bxx},
  MRNUMBER = {96a:52011},
MRREVIEWER = {Margaret M. Bayer},
}

@book{ccd,
    AUTHOR = {Beck, M. and Robins, S.},
     TITLE = {Computing the {C}ontinuous {D}iscretely: Integer-Point {E}numeration in
{P}olyhedra},
    SERIES = {Undergraduate Texts in Mathematics},
   EDITION = {Second Edition},
 PUBLISHER = {Springer, New York},
      YEAR = {2015},
url = {Electronically available at \url{http://math.sfsu.edu/beck/ccd.html}},
     PAGES = {xx+285},
   MRCLASS = {11P21 (05A15 05B15 11-02 11H06 52B05 52B20)},
}
@book{crt,
	series = {Graduate Studies in Mathematics},
	title = {Combinatorial Reciprocity Theorems: An Invitation to Enumerative Geometric Combinatorics},
	isbn = {978-1-4704-2200-4},
	shorttitle = {Combinatorial Reciprocity Theorems},
	number = {195},
	publisher = {Amer. Math. Soc., Providence, RI},
	author = {Beck, M. and Sanyal, R.},
	year = {2018},
}

@incollection {batryrevborisov,
    AUTHOR = {Batyrev,  V. V. and Borisov, L. A.},
     TITLE = {Dual cones and mirror symmetry for generalized {C}alabi--{Y}au manifolds},
 BOOKTITLE = {Mirror Symmetry, {II}},
    SERIES = {AMS/IP Stud. Adv. Math.},
    VOLUME = {1},
     PAGES = {71--86},
 PUBLISHER = {Amer. Math. Soc., Providence, RI},
      YEAR = {1997},
   MRCLASS = {14J32 (14M25)},
 MRNUMBER = {1416334},
MRREVIEWER = {Dimitrios I. Dais},
}

@article {brunsroemer,
    AUTHOR = {Bruns, W. and R{\"o}mer, T.},
     TITLE = {{$h$}-vectors of {G}orenstein polytopes},
   JOURNAL = {J. Combin. Theory Ser. A},
  FJOURNAL = {Journal of Combinatorial Theory. Series A},
    VOLUME = {114},
      YEAR = {2007},
    NUMBER = {1},
     PAGES = {65--76},
      ISSN = {0097-3165},
     CODEN = {JCBTA7},
   MRCLASS = {05E25 (13F55 52B20)},
 MRNUMBER = {2275581 (2007m:05230)},
MRREVIEWER = {Hugh Ross Thomas},
}

@article {linke,
    AUTHOR = {Linke, E.},
     TITLE = {Rational {E}hrhart quasi-polynomials},
   JOURNAL = {J. Combin. Theory Ser. A},
  FJOURNAL = {Journal of Combinatorial Theory. Series A},
    VOLUME = {118},
      YEAR = {2011},
    NUMBER = {7},
     PAGES = {1966--1978},
      ISSN = {0097-3165},
     CODEN = {JCBTA7},
   MRCLASS = {52B20 (52C07)},
 MRNUMBER = {2802181},
MRREVIEWER = {Benjamin Braun},
}

@article {stanleyhilbertgradedalgebras,
    AUTHOR = {Stanley, R. P.},
     TITLE = {Hilbert functions of graded algebras},
   JOURNAL = {Adv. Math.},
  FJOURNAL = {Advances in Mathematics},
    VOLUME = {28},
      YEAR = {1978},
    NUMBER = {1},
     PAGES = {57--83},
      ISSN = {0001-8708},
   MRCLASS = {13D10 (13H10)},
 MRNUMBER = {0485835 (58 \#5637)},
MRREVIEWER = {Idun Reiten},
}

@article {hibidual,
    AUTHOR = {Hibi, T.},
     TITLE = {Dual polytopes of rational convex polytopes},
   JOURNAL = {Combinatorica},
  FJOURNAL = {Combinatorica. An International Journal on Combinatorics and
              the Theory of Computing},
    VOLUME = {12},
      YEAR = {1992},
    NUMBER = {2},
     PAGES = {237--240},
      ISSN = {0209-9683},
     CODEN = {COMBDI},
   MRCLASS = {52B20},
  MRNUMBER = {MR1179260 (93f:52018)},
MRREVIEWER = {P. McMullen},
}

@article {kasprzyknill,
    AUTHOR = {Kasprzyk, A. M. and Nill, B.},
     TITLE = {Reflexive polytopes of higher index and the number 12},
   JOURNAL = {Electron. J. Combin.},
  FJOURNAL = {Electronic Journal of Combinatorics},
    VOLUME = {19},
      YEAR = {2012},
    NUMBER = {3},
     PAGES = {Paper 9, 18 pp.},
   MRCLASS = {52B20 (14M25 14Q10)},
 MRNUMBER = {2967214},
MRREVIEWER = {Ivan Arzhantsev},
}

@article {fisetkasprzyk,
    AUTHOR = {Fiset, M. H. J. and Kasprzyk, A. M.},
     TITLE = {A note on palindromic {$\delta$}-vectors for certain rational
              polytopes},
   JOURNAL = {Electron. J. Combin.},
  FJOURNAL = {Electronic Journal of Combinatorics},
    VOLUME = {15},
      YEAR = {2008},
    NUMBER = {1},
     PAGES = {Note 18, 4 pp.},
      ISSN = {1077-8926},
   MRCLASS = {05A15 (11H06)},
  MRNUMBER = {2411464 (2009e:05017)},
MRREVIEWER = {Matthias Beck},
}

@article {stapledonweightedehrart,
    AUTHOR = {Stapledon, A.},
     TITLE = {Weighted {E}hrhart theory and orbifold cohomology},
   JOURNAL = {Adv. Math.},
  FJOURNAL = {Advances in Mathematics},
    VOLUME = {219},
      YEAR = {2008},
    NUMBER = {1},
     PAGES = {63--88},
      ISSN = {0001-8708},
     CODEN = {ADMTA4},
   MRCLASS = {14M25 (14F43 52B05)},
 MRNUMBER = {2435420 (2009i:14074)},
MRREVIEWER = {J. Maurice Rojas},
}

@article {batyrevdualpoly,
    AUTHOR = {Batyrev, V. V.},
     TITLE = {Dual polyhedra and mirror symmetry for {C}alabi--{Y}au
              hypersurfaces in toric varieties},
   JOURNAL = {J. Algebraic Geom.},
  FJOURNAL = {Journal of Algebraic Geometry},
    VOLUME = {3},
      YEAR = {1994},
    NUMBER = {3},
     PAGES = {493--535},
      ISSN = {1056-3911},
   MRCLASS = {14J32 (14M25)},
  MRNUMBER = {MR1269718 (95c:14046)},
MRREVIEWER = {I. Dolgachev},
}

@misc{beck2021rational,
      title={Rational Ehrhart theory}, 
      author={M. Beck and S. Elia and S. Rehberg},
      year={2021},
      eprint={2110.10204},
      EPRINTTYPE = {arxiv},
      primaryClass={math.CO}
}