{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,22]],"date-time":"2026-08-22T01:35:02Z","timestamp":1787362502734,"version":"build-2736575974"},"reference-count":7,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Discrete Math."],"published-print":{"date-parts":[[2021,1]]},"abstract":"<jats:p>Let $a_1,\\ldots,a_n$ and $b_1,\\ldots,b_m$ be fixed positive integers, and let ${\\mathcal S}$ denote the set of all nonnegative integer solutions of the equation $x_1a_1+\\cdots +x_na_n=y_1b_1+\\cdots +y_mb_m$. A solution $(x_1,\\ldots,x_n,y_1,\\ldots,y_m)$ in ${\\mathcal S}$ is called minimal if it cannot be expressed as the sum of two nonzero solutions in ${\\mathcal S}$. For each pair $(i,j)$ with $1\\leq i\\leq n$ and $1\\leq j\\leq m$, the solution whose only nonzero coordinates are $x_i=b_j$ and $y_j=a_i$ is called a generator. Our main result shows that every minimal solution is a convex combination of the generators and the zero-solution. This proves a conjecture of Henk--Weismantel and, independently, Hosten--Sturmfels.<\/jats:p>","DOI":"10.1137\/21m1404417","type":"journal-article","created":{"date-parts":[[2021,8,30]],"date-time":"2021-08-30T17:00:00Z","timestamp":1630342800000},"page":"1952-1963","source":"Crossref","is-referenced-by-count":2,"title":["Geometry of the Minimal Solutions of a Linear Diophantine Equation"],"prefix":"10.1137","volume":"35","author":[{"given":"Papa A.","family":"Sissokho","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2021,8,30]]},"reference":[{"key":"atypb1","first-page":"1","author":"Diaconis P.","year":"1995","journal-title":"Budapest"},{"key":"atypb2","doi-asserted-by":"crossref","unstructured":"J. De Loera, R. Hemmecke, and M. K\u00f6ppe,\n                      Algebraic and Geometric Ideas in the Theory of Discrete Optimization\n                      , MOS-SIAM Series on Optimization 14, SIAM, Philadelphia, 2013.","DOI":"10.1137\/1.9781611972443"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1007\/BF01681344"},{"key":"atypb4","first-page":"49","volume":"41","author":"Henk M.","year":"2000","journal-title":"Contrib. Algebra Geom."},{"key":"atypb5","first-page":"39","volume":"305","author":"Lambert J.","year":"1987","journal-title":"C. R. Acad. Sci. Paris Ser. I Math."},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.4064\/aa166-3-4"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1215\/S0012-7094-73-04054-4"}],"container-title":["SIAM Journal on Discrete Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/21M1404417","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:07:08Z","timestamp":1787321228000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/21M1404417"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,1]]},"references-count":7,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2021,1]]}},"alternative-id":["10.1137\/21M1404417"],"URL":"https:\/\/doi.org\/10.1137\/21m1404417","relation":{},"ISSN":["0895-4801","1095-7146"],"issn-type":[{"value":"0895-4801","type":"print"},{"value":"1095-7146","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,1]]}}}