{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,23]],"date-time":"2026-04-23T20:44:41Z","timestamp":1776977081556,"version":"3.51.4"},"reference-count":27,"publisher":"American Mathematical Society (AMS)","issue":"255","license":[{"start":{"date-parts":[[2007,3,10]],"date-time":"2007-03-10T00:00:00Z","timestamp":1173484800000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    We present a polynomial time algorithm to compute any fixed number of the highest coefficients of the Ehrhart quasi-polynomial of a rational simplex. Previously such algorithms were known for integer simplices and for rational polytopes of a fixed dimension. The algorithm is based on the formula relating the\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"k\">\n                        <mml:semantics>\n                          <mml:mi>k<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">k<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    th coefficient of the Ehrhart quasi-polynomial of a rational polytope to volumes of sections of the polytope by affine lattice subspaces parallel to\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"k\">\n                        <mml:semantics>\n                          <mml:mi>k<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">k<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -dimensional faces of the polytope. We discuss possible extensions and open questions.\n                  <\/p>","DOI":"10.1090\/s0025-5718-06-01836-9","type":"journal-article","created":{"date-parts":[[2006,5,24]],"date-time":"2006-05-24T14:43:01Z","timestamp":1148481781000},"page":"1449-1466","source":"Crossref","is-referenced-by-count":29,"title":["Computing the Ehrhart quasi-polynomial of a rational simplex"],"prefix":"10.1090","volume":"75","author":[{"given":"Alexander","family":"Barvinok","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2006,3,10]]},"reference":[{"issue":"4","key":"1","doi-asserted-by":"publisher","first-page":"625","DOI":"10.1007\/BF01445125","article-title":"New bounds in some transference theorems in the geometry of numbers","volume":"296","author":"Banaszczyk, W.","year":"1993","journal-title":"Math. Ann.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5831","issn-type":"print"},{"key":"2","unstructured":"A. Barvinok, Computing the Ehrhart polynomial of a convex lattice polytope, preprint TRITA-MAT-1992-0036 (1992), Royal Institute of Technology, Stockholm."},{"issue":"2","key":"3","doi-asserted-by":"publisher","first-page":"123","DOI":"10.1007\/BF02573970","article-title":"Computing the volume, counting integral points, and exponential sums","volume":"10","author":"Barvinok, Alexander I.","year":"1993","journal-title":"Discrete Comput. Geom.","ISSN":"https:\/\/id.crossref.org\/issn\/0179-5376","issn-type":"print"},{"issue":"4","key":"4","doi-asserted-by":"publisher","first-page":"769","DOI":"10.1287\/moor.19.4.769","article-title":"A polynomial time algorithm for counting integral points in polyhedra when the dimension is fixed","volume":"19","author":"Barvinok, Alexander I.","year":"1994","journal-title":"Math. Oper. 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