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Our tools are algebraic and number theoretic in nature and include Siegel's lemma, generating functions, and commutative algebra. These results have some interesting consequences in discrete optimization.<\/jats:p>","DOI":"10.1137\/16m1083876","type":"journal-article","created":{"date-parts":[[2017,4,4]],"date-time":"2017-04-04T17:50:36Z","timestamp":1491328236000},"page":"239-253","source":"Crossref","is-referenced-by-count":29,"title":["Sparse Solutions of Linear Diophantine Equations"],"prefix":"10.1137","volume":"1","author":[{"given":"Iskander","family":"Aliev","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jes\u00fas A.","family":"De Loera","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Timm","family":"Oertel","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Christopher","family":"O'Neill","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2017,4,4]]},"reference":[{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1007\/s00454-008-9059-9"},{"key":"atypb3","first-page":"339","author":"Alon N.","year":"2009","journal-title":"Berlin"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1007\/s00233-015-9740-2"},{"key":"atypb5","first-page":"91","author":"Barvinok A. 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Micciancio,\n                      Locally dense codes\n                      , Proceedings of the 29th IEEE Conference on Computational Complexity-CC 2014, IEEE Computer Society, Los Alamitos, CA, 2014, pp. 90-97,https:\/\/doi.org\/10.1109\/CCC.2014.17.","DOI":"10.1109\/CCC.2014.17"},{"key":"atypb21","unstructured":"E. Miller and B. Sturmfels,\n                      Combinatorial Commutative Algebra\n                      , Grad. Texts Math. 227, Springer-Verlag, New York, 2005."},{"key":"atypb23","doi-asserted-by":"crossref","unstructured":"J. L. Rami\u0301rez Alfonsi\u0301n,\n                      The Diophantine Frobenius Problem\n                      , Oxford Lecture Ser. Math. Appl. 30, Oxford University Press, New York, 2005,https:\/\/doi.org\/10.1093\/acprof:oso\/9780198568209.001.0001.","DOI":"10.1093\/acprof:oso\/9780198568209.001.0001"},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1007\/s00605-002-0491-2"},{"key":"atypb25","unstructured":"A. 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