{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,13]],"date-time":"2026-05-13T21:43:38Z","timestamp":1778708618834,"version":"3.51.4"},"reference-count":17,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2015,10,13]],"date-time":"2015-10-13T00:00:00Z","timestamp":1444694400000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Discrete Comput Geom"],"published-print":{"date-parts":[[2015,12]]},"DOI":"10.1007\/s00454-015-9734-6","type":"journal-article","created":{"date-parts":[[2015,10,13]],"date-time":"2015-10-13T18:38:49Z","timestamp":1444761529000},"page":"930-953","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["On Triple Intersections of Three Families of Unit Circles"],"prefix":"10.1007","volume":"54","author":[{"given":"Orit E.","family":"Raz","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Micha","family":"Sharir","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"J\u00f3zsef","family":"Solymosi","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2015,10,13]]},"reference":[{"key":"9734_CR1","volume-title":"Using Algebraic Geometry","author":"DA Cox","year":"2005","unstructured":"Cox, D.A., Little, J., O\u2019Shea, D.: Using Algebraic Geometry, 2nd edn. Springer, Heidelberg (2005)","edition":"2"},{"key":"9734_CR2","unstructured":"Elekes, G.: Sums versus products in number theory, algebra and Erd\u0151s geometry\u2014a survey. In: Paul Erd\u0151s and His Mathematics II, vol. 11, pp.\u00a0241\u2013290. Bolyai Mathematical Society Studies, Budapest (2002)"},{"key":"9734_CR3","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1006\/jcta.1999.2976","volume":"89","author":"G Elekes","year":"2000","unstructured":"Elekes, G., R\u00f3nyai, L.: A combinatorial problem on polynomials and rational functions. J. Comb. Theory, Ser. A 89, 1\u201320 (2000)","journal-title":"J. Comb. Theory, Ser. A"},{"key":"9734_CR4","doi-asserted-by":"crossref","first-page":"691","DOI":"10.1017\/S0963548309990265","volume":"18","author":"G Elekes","year":"2009","unstructured":"Elekes, G., Simonovits, M., Szab\u00f3, E.: A combinatorial distinction between unit circles and straight lines: how many coincidences can they have? Comb. Probab. Comput. 18, 691\u2013705 (2009)","journal-title":"Comb. Probab. Comput."},{"key":"9734_CR5","doi-asserted-by":"crossref","first-page":"537","DOI":"10.1007\/s00493-012-2505-6","volume":"32","author":"G Elekes","year":"2012","unstructured":"Elekes, G., Szab\u00f3, E.: How to find groups? (And how to use them in Erd\u0151s geometry?). Combinatorica 32, 537\u2013571 (2012)","journal-title":"Combinatorica"},{"issue":"2","key":"9734_CR6","doi-asserted-by":"crossref","first-page":"275","DOI":"10.1090\/S0002-9939-1964-0161339-9","volume":"15","author":"J Milnor","year":"1964","unstructured":"Milnor, J.: On the Betti numbers of real varieties. Proc. Am. Math. Soc. 15(2), 275\u2013280 (1964)","journal-title":"Proc. Am. Math. Soc."},{"key":"9734_CR7","doi-asserted-by":"crossref","DOI":"10.1002\/9781118033203","volume-title":"Combinatorial Geometry","author":"J Pach","year":"1995","unstructured":"Pach, J., Agarwal, P.K.: Combinatorial Geometry. Wiley-Interscience, New York (1995)"},{"key":"9734_CR8","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1017\/S0963548397003192","volume":"7","author":"J Pach","year":"1998","unstructured":"Pach, J., Sharir, M.: On the number of incidences between points and curves. Comb. Probab. Comput. 7, 121\u2013127 (1998)","journal-title":"Comb. Probab. Comput."},{"key":"9734_CR9","unstructured":"Raz, O.E., Sharir, M.: Unit-area triangles: theme and variations. In: Proceedings of the 31st Annual Symposium on Computational Geometry, pp. 569\u2013583 (2015). http:\/\/arxiv.org\/abs\/1501.00379"},{"key":"9734_CR10","unstructured":"Raz, O.E., Sharir, M., Solymosi, J.: Polynomials vanishing on grids: the Elekes\u2013R\u00f3nyai problem revisited. Am. J. Math. (2015). http:\/\/arxiv.org\/abs\/1401.7419"},{"key":"9734_CR11","unstructured":"Raz, O.E., Sharir, M., de Zeeuw, F.: Polynomials vanishing on Cartesian products: the Elekes\u2013Szab\u00f3 Theorem revisited. In: Proceedings of the 31st Annual Symposium on Computational Geometry, pp. 522\u2013536 (2015). http:\/\/arxiv.org\/abs\/1504.05012"},{"issue":"4","key":"9734_CR12","doi-asserted-by":"crossref","first-page":"701","DOI":"10.1145\/322217.322225","volume":"27","author":"J Schwartz","year":"1980","unstructured":"Schwartz, J.: Fast probabilistic algorithms for verification of polynomial identities. J. ACM 27(4), 701\u2013717 (1980)","journal-title":"J. ACM"},{"key":"9734_CR13","doi-asserted-by":"crossref","first-page":"1732","DOI":"10.1016\/j.jcta.2013.06.009","volume":"20","author":"M Sharir","year":"2013","unstructured":"Sharir, M., Sheffer, A., Solymosi, J.: Distinct distances on two lines. J. Comb. Theory, Ser. A 20, 1732\u20131736 (2013)","journal-title":"J. Comb. Theory, Ser. A"},{"key":"9734_CR14","unstructured":"Sharir, M., Solymosi, J.: Distinct distances from three points. http:\/\/arxiv.org\/abs\/1308.0814"},{"key":"9734_CR15","doi-asserted-by":"crossref","first-page":"353","DOI":"10.1017\/S0963548397002976","volume":"6","author":"L Sz\u00e9kely","year":"1997","unstructured":"Sz\u00e9kely, L.: Crossing numbers and hard Erd\u0151s problems in discrete geometry. Comb. Probab. Comput. 6, 353\u2013358 (1997)","journal-title":"Comb. Probab. Comput."},{"key":"9734_CR16","doi-asserted-by":"crossref","first-page":"255","DOI":"10.1515\/9781400874842-016","volume-title":"Differential and Combinatorial Topology","author":"R Thom","year":"1965","unstructured":"Thom, R.: Sur l\u2019homologie des vari\u00e9t\u00e9s algebriques r\u00e9elles. In: Cairns, S.S. (ed.) Differential and Combinatorial Topology, pp. 255\u2013265. Princeton University Press, Princeton (1965)"},{"issue":"4","key":"9734_CR17","doi-asserted-by":"crossref","first-page":"207","DOI":"10.1016\/0020-0190(89)90143-9","volume":"33","author":"R Zippel","year":"1989","unstructured":"Zippel, R.: An explicit separation of relativised random polynomial time and relativised deterministic polynomial time. Inf. Process. Lett. 33(4), 207\u2013212 (1989)","journal-title":"Inf. Process. Lett."}],"container-title":["Discrete & Computational Geometry"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00454-015-9734-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s00454-015-9734-6\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00454-015-9734-6","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,31]],"date-time":"2019-08-31T13:45:29Z","timestamp":1567259129000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s00454-015-9734-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,10,13]]},"references-count":17,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2015,12]]}},"alternative-id":["9734"],"URL":"https:\/\/doi.org\/10.1007\/s00454-015-9734-6","relation":{},"ISSN":["0179-5376","1432-0444"],"issn-type":[{"value":"0179-5376","type":"print"},{"value":"1432-0444","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015,10,13]]}}}