{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,13]],"date-time":"2026-05-13T21:43:50Z","timestamp":1778708630845,"version":"3.51.4"},"reference-count":22,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2016,7,15]],"date-time":"2016-07-15T00:00:00Z","timestamp":1468540800000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2017,1]]},"abstract":"<jats:p>Let<jats:italic>S<\/jats:italic>be a set of<jats:italic>n<\/jats:italic>points in<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0963548316000225_inline1\"\/><jats:tex-math>${\\mathbb R}^{2}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>contained in an algebraic curve<jats:italic>C<\/jats:italic>of degree<jats:italic>d<\/jats:italic>. We prove that the number of distinct distances determined by<jats:italic>S<\/jats:italic>is at least<jats:italic>c<jats:sub>d<\/jats:sub>n<\/jats:italic><jats:sup>4\/3<\/jats:sup>, unless<jats:italic>C<\/jats:italic>contains a line or a circle.<\/jats:p><jats:p>We also prove the lower bound<jats:italic>c<jats:sub>d<\/jats:sub><\/jats:italic>\u2032 min{<jats:italic>m<\/jats:italic><jats:sup>2\/3<\/jats:sup><jats:italic>n<\/jats:italic><jats:sup>2\/3<\/jats:sup>,<jats:italic>m<\/jats:italic><jats:sup>2<\/jats:sup>,<jats:italic>n<\/jats:italic><jats:sup>2<\/jats:sup>} for the number of distinct distances between<jats:italic>m<\/jats:italic>points on one irreducible plane algebraic curve and<jats:italic>n<\/jats:italic>points on another, unless the two curves are parallel lines, orthogonal lines, or concentric circles. This generalizes a result on distances between lines of Sharir, Sheffer and Solymosi in [19].<\/jats:p>","DOI":"10.1017\/s0963548316000225","type":"journal-article","created":{"date-parts":[[2016,7,15]],"date-time":"2016-07-15T09:58:40Z","timestamp":1468576720000},"page":"99-117","source":"Crossref","is-referenced-by-count":10,"title":["Distinct Distances on Algebraic Curves in the Plane"],"prefix":"10.1017","volume":"26","author":[{"given":"J\u00c1NOS","family":"PACH","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"FRANK","family":"DE ZEEUW","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2016,7,15]]},"reference":[{"key":"S0963548316000225_ref10","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781139173285"},{"key":"S0963548316000225_ref3","volume-title":"Research Problems in Discrete Geometry","author":"Brass","year":"2005"},{"key":"S0963548316000225_ref2","doi-asserted-by":"crossref","DOI":"10.1007\/3-540-33099-2","volume-title":"Algorithms in Real Algebraic Geometry","author":"Basu","year":"2006"},{"key":"S0963548316000225_ref21","article-title":"Few distinct distances implies no heavy lines or circles","author":"Sheffer","year":"2014","journal-title":"Combinatorica"},{"key":"S0963548316000225_ref19","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcta.2013.06.009"},{"key":"S0963548316000225_ref11","doi-asserted-by":"publisher","DOI":"10.4007\/annals.2015.181.1.2"},{"key":"S0963548316000225_ref18","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcta.2013.06.004"},{"key":"S0963548316000225_ref6","doi-asserted-by":"publisher","DOI":"10.1023\/A:1004802524095"},{"key":"S0963548316000225_ref12","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-2189-8"},{"key":"S0963548316000225_ref15","doi-asserted-by":"publisher","DOI":"10.1016\/0097-3165(92)90094-B"},{"key":"S0963548316000225_ref14","doi-asserted-by":"publisher","DOI":"10.1007\/s00454-012-9443-3"},{"key":"S0963548316000225_ref8","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548311000137"},{"key":"S0963548316000225_ref9","doi-asserted-by":"publisher","DOI":"10.2307\/2305092"},{"key":"S0963548316000225_ref16","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548397003192"},{"key":"S0963548316000225_ref17","doi-asserted-by":"crossref","unstructured":"Raz O. E. , Sharir M. and Solymosi J. (2014) Polynomials vanishing on grids: The Elekes\u2013R\u00f3nyai problem revisited. In SOCG'14: Proceedings of the 30th Annual Symposium on Computational Geometry, pp. 198\u2013205.","DOI":"10.1145\/2582112.2582149"},{"key":"S0963548316000225_ref5","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-387-35651-8"},{"key":"S0963548316000225_ref1","doi-asserted-by":"publisher","DOI":"10.1112\/plms\/pdv059"},{"key":"S0963548316000225_ref13","doi-asserted-by":"publisher","DOI":"10.1016\/0304-3975(83)90002-6"},{"key":"S0963548316000225_ref22","doi-asserted-by":"publisher","DOI":"10.1007\/s00454-012-9420-x"},{"key":"S0963548316000225_ref4","doi-asserted-by":"publisher","DOI":"10.1007\/s00454-014-9586-5"},{"key":"S0963548316000225_ref7","doi-asserted-by":"publisher","DOI":"10.1006\/jcta.1999.2976"},{"key":"S0963548316000225_ref20","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548315000231"}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548316000225","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,9,11]],"date-time":"2019-09-11T05:45:52Z","timestamp":1568180752000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548316000225\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,7,15]]},"references-count":22,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2017,1]]}},"alternative-id":["S0963548316000225"],"URL":"https:\/\/doi.org\/10.1017\/s0963548316000225","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2016,7,15]]}}}