{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,9]],"date-time":"2026-05-09T02:40:03Z","timestamp":1778294403465,"version":"3.51.4"},"reference-count":29,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2024,10,5]],"date-time":"2024-10-05T00:00:00Z","timestamp":1728086400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2024,10,5]],"date-time":"2024-10-05T00:00:00Z","timestamp":1728086400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/100000143","name":"Division of Computing and Communication Foundations","doi-asserted-by":"publisher","award":["2331400"],"award-info":[{"award-number":["2331400"]}],"id":[{"id":"10.13039\/100000143","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000143","name":"Division of Computing and Communication Foundations","doi-asserted-by":"publisher","award":["2331401"],"award-info":[{"award-number":["2331401"]}],"id":[{"id":"10.13039\/100000143","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Robert and Sara Lumpkins Collegiate Professorship"},{"DOI":"10.13039\/100000893","name":"Simons Foundation","doi-asserted-by":"publisher","award":["SFM-00005696"],"award-info":[{"award-number":["SFM-00005696"]}],"id":[{"id":"10.13039\/100000893","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000893","name":"Simons Foundation","doi-asserted-by":"publisher","award":["965262"],"award-info":[{"award-number":["965262"]}],"id":[{"id":"10.13039\/100000893","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2025,9]]},"abstract":"<jats:title>Abstract<\/jats:title>\n          <jats:p>A standard question in real algebraic geometry is to compute the number of connected components of a real algebraic variety in affine space. This manuscript provides algorithms for computing the number of connected components, the Euler characteristic, and deciding the connectivity between two points for a smooth manifold arising as the complement of a real hypersurface of a real algebraic variety. When considering the complement of the set of singular points of a real algebraic variety, this yields an approach for determining smooth connectivity in a real algebraic variety. The method is based upon gradient ascent\/descent paths on the real algebraic variety inspired by a method proposed by Hong, Rohal, Safey El Din, and Schost for complements of real hypersurfaces. Several examples are included to demonstrate\u00a0the\u00a0approach.<\/jats:p>","DOI":"10.1007\/s11075-024-01952-3","type":"journal-article","created":{"date-parts":[[2024,10,5]],"date-time":"2024-10-05T00:01:32Z","timestamp":1728086492000},"page":"63-84","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Smooth connectivity in real algebraic varieties"],"prefix":"10.1007","volume":"100","author":[{"given":"Joseph","family":"Cummings","sequence":"first","affiliation":[]},{"given":"Jonathan D.","family":"Hauenstein","sequence":"additional","affiliation":[]},{"given":"Hoon","family":"Hong","sequence":"additional","affiliation":[]},{"given":"Clifford D.","family":"Smyth","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2024,10,5]]},"reference":[{"issue":"1","key":"1952_CR1","first-page":"1","volume":"18","author":"A Ambrosetti","year":"1984","unstructured":"Ambrosetti, A.: Elliptic equations with jumping nonlinearities. J. Math. Phys. Sci. 18(1), 1\u201312 (1984)","journal-title":"J. Math. Phys. Sci."},{"key":"1952_CR2","doi-asserted-by":"crossref","unstructured":"Banyaga, A., Hurtubise, D.: Lectures on Morse homology. Kluwer Texts in the Mathematical Sciences, vol. 29. Kluwer Academic Publishers Group, Dordrecht (2004)","DOI":"10.1007\/978-1-4020-2696-6"},{"issue":"1","key":"1952_CR3","doi-asserted-by":"publisher","first-page":"55","DOI":"10.1090\/S0894-0347-99-00311-2","volume":"13","author":"S Basu","year":"2000","unstructured":"Basu, S., Pollack, R., Roy, M.-F.: Computing roadmaps of semi-algebraic sets on a variety. J. Amer. Math. Soc. 13(1), 55\u201382 (2000)","journal-title":"J. Amer. Math. Soc."},{"key":"1952_CR4","doi-asserted-by":"crossref","unstructured":"Basu, S., Pollack, R., Roy, M.-F.: Algorithms in real algebraic geometry, volume\u00a010 of Algorithms and Computation in Mathematics. Springer-Verlag, Berlin, second edition, (2006)","DOI":"10.1007\/3-540-33099-2"},{"issue":"2","key":"1952_CR5","doi-asserted-by":"publisher","first-page":"278","DOI":"10.1007\/s00454-014-9610-9","volume":"52","author":"S Basu","year":"2014","unstructured":"Basu, S., Roy, M.-F.: Divide and conquer roadmap for algebraic sets. Discrete Comput. Geom. 52(2), 278\u2013343 (2014)","journal-title":"Discrete Comput. Geom."},{"issue":"6","key":"1952_CR6","doi-asserted-by":"publisher","first-page":"1117","DOI":"10.1007\/s10208-014-9212-1","volume":"14","author":"S Basu","year":"2014","unstructured":"Basu, S., Roy, M.-F., Safey El Din, M., Schost, E.: A baby step-giant step roadmap algorithm for general algebraic sets. Found. Comput. Math. 14(6), 1117\u20131172 (2014)","journal-title":"Found. Comput. Math."},{"key":"1952_CR7","first-page":"55","volume-title":"Approximate Commutative Algebra","author":"DJ Bates","year":"2010","unstructured":"Bates, D.J., Hauenstein, J.D., Peterson, C., Sommese, A.J.: Numerical decomposition of the rank-deficiency set of a matrix of multivariate polynomials. In: Robbiano, L., Abbott, J. (eds.) Approximate Commutative Algebra, pp. 55\u201377. Springer Vienna, Vienna (2010)"},{"key":"1952_CR8","doi-asserted-by":"publisher","unstructured":"Bates, D.J., Hauenstein, J.D., Sommese, A.J., Wampler, C.W.: Bertini: Software for numerical algebraic geometry, (2006). Available at bertini.nd.edu with permanent https:\/\/doi.org\/10.7274\/R0H41PB5","DOI":"10.7274\/R0H41PB5"},{"key":"1952_CR9","doi-asserted-by":"crossref","unstructured":"Becker, E., Neuhaus, R.: Computation of real radicals of polynomial ideals. In: Computational algebraic geometry (Nice, 1992), volume 109 of Progr. Math., pp. 1\u201320. Birkh\u00e4user Boston, Boston, MA, (1993)","DOI":"10.1007\/978-1-4612-2752-6_1"},{"key":"1952_CR10","doi-asserted-by":"crossref","unstructured":"Bohigas, O., Henderson, M.E., Ros, L., Porta, J.M.: A singularity-free path planner for closed-chain manipulators. In: 2012 IEEE International Conference on Robotics and Automation, pp. 2128\u20132134, (2012)","DOI":"10.1109\/ICRA.2012.6224899"},{"key":"1952_CR11","doi-asserted-by":"crossref","unstructured":"Bohigas, O., Manubens, M., Ros, L.: Singularities of Robot Mechanisms: Numerical Computation and Avoidance Path Planning, volume\u00a041 of Mechanisms and Machine Science. Springer, [Cham], (2017)","DOI":"10.1007\/978-3-319-32922-2"},{"issue":"12","key":"1952_CR12","doi-asserted-by":"publisher","first-page":"5232","DOI":"10.1016\/j.jpaa.2019.03.019","volume":"223","author":"DA Brake","year":"2019","unstructured":"Brake, D.A., Hauenstein, J.D., Vinzant, C.: Computing complex and real tropical curves using monodromy. J. Pure Appl. Algebra 223(12), 5232\u20135250 (2019)","journal-title":"J. Pure Appl. Algebra"},{"key":"1952_CR13","unstructured":"Canny, J.: The complexity of robot motion planning. ACM Doctoral Dissertation Awards, vol. 1987. MIT Press, Cambridge, MA (1988)"},{"issue":"5","key":"1952_CR14","doi-asserted-by":"publisher","first-page":"504","DOI":"10.1093\/comjnl\/36.5.504","volume":"36","author":"J Canny","year":"1993","unstructured":"Canny, J.: Computing roadmaps of general semi-algebraic sets. Comput. J. 36(5), 504\u2013514 (1993)","journal-title":"Comput. J."},{"issue":"4","key":"1952_CR15","doi-asserted-by":"publisher","first-page":"217","DOI":"10.1007\/BF01614146","volume":"2","author":"J Canny","year":"1992","unstructured":"Canny, J., Grigor\u2019ev, D.Y., Vorobjov, N.N., Jr.: Finding connected components of a semialgebraic set in subexponential time. Appl. Algebra Engrg. Comm. Comput. 2(4), 217\u2013238 (1992)","journal-title":"Appl. Algebra Engrg. Comm. Comput."},{"key":"1952_CR16","doi-asserted-by":"crossref","unstructured":"Edwards, P.B., Baskar, A., Hills, C., Plecnik, M., Hauenstein, J.D.: Output mode switching for parallel five-bar manipulators using a graph-based path planner. In: 2023 IEEE International Conference on Robotics and Automation (ICRA), pp. 9735\u20139741, (2023)","DOI":"10.1109\/ICRA48891.2023.10160891"},{"issue":"2","key":"1952_CR17","doi-asserted-by":"publisher","first-page":"133","DOI":"10.1007\/BF01202001","volume":"2","author":"DY Grigor\u2019ev","year":"1992","unstructured":"Grigor\u2019ev, D.Y., Vorobjov, N.N., Jr.: Counting connected components of a semialgebraic set in subexponential time. Comput. Complexity 2(2), 133\u2013186 (1992)","journal-title":"Comput. Complexity"},{"key":"1952_CR18","doi-asserted-by":"crossref","unstructured":"Heintz, J., Roy, M.-F., Solern\u00f3, P.: Single exponential path finding in semialgebraic sets. I. The case of a regular bounded hypersurface. In: Applied algebra, algebraic algorithms and error-correcting codes (Tokyo, 1990), volume 508 of Lecture Notes in Comput. Sci., pp. 180\u2013196. Springer, Berlin, (1991)","DOI":"10.1007\/3-540-54195-0_50"},{"key":"1952_CR19","first-page":"449","volume-title":"Algebraic geometry and its applications (West Lafayette IN, 1990)","author":"J Heintz","year":"1994","unstructured":"Heintz, J., Roy, M.-F., Solern\u00f3, P.: Single exponential path finding in semi-algebraic sets. II. The general case. In: Algebraic geometry and its applications (West Lafayette IN, 1990), pp. 449\u2013465. Springer, New York (1994)"},{"key":"1952_CR20","doi-asserted-by":"crossref","unstructured":"Hong, H.: Connectivity in semi-algebraic sets. In: T. Ida, V. Negru, T. Jebelean, D. Petcu, S. M. Watt, D. Zaharie, (Eds.), 12th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing, SYNASC 2010, Timisoara, Romania, 23-26 September 2010, pp. 4\u20137. IEEE Computer Society, (2010)","DOI":"10.1109\/SYNASC.2010.91"},{"key":"1952_CR21","unstructured":"Hong, H., Rohal, J., Safey El\u00a0Din, M., Schost, E.: Connectivity in semi-algebraic sets I. arXiv:2011.02162, (2020)"},{"key":"1952_CR22","volume-title":"Numerical Analysis: Mathematics of Scientific Computing","author":"D Kincaid","year":"2002","unstructured":"Kincaid, D., Cheney, W.: Numerical Analysis: Mathematics of Scientific Computing, 3rd edn. Brooks\/Cole Publishing Co., Pacific Grove, CA (2002)","edition":"3"},{"key":"1952_CR23","doi-asserted-by":"crossref","unstructured":"Kummer, M., Sturmfels, B., Vlad, R.: Maximal Mumford curves from planar graphs. arXiv:2404.11838, (2024)","DOI":"10.4310\/PAMQ.250402030514"},{"issue":"11","key":"1952_CR24","doi-asserted-by":"publisher","first-page":"620","DOI":"10.1287\/mnsc.18.11.620","volume":"18","author":"DG Luenberger","year":"1972","unstructured":"Luenberger, D.G.: The gradient projection method along geodesics. Manag. Sci. 18(11), 620\u2013631 (1972)","journal-title":"Manag. Sci."},{"issue":"1","key":"1952_CR25","doi-asserted-by":"publisher","first-page":"151","DOI":"10.1007\/s10107-003-0489-0","volume":"100","author":"JJ Mor\u00e9","year":"2004","unstructured":"Mor\u00e9, J.J., Munson, T.S.: Computing mountain passes and transition states. Math. Program. 100(1), 151\u2013182 (2004)","journal-title":"Math. Program."},{"key":"1952_CR26","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4614-1105-5","volume-title":"An Invitation to Morse Theory","author":"L Nicolaescu","year":"2011","unstructured":"Nicolaescu, L.: An Invitation to Morse Theory, 2nd edn. Universitext. Springer, New York (2011)","edition":"2"},{"key":"1952_CR27","volume-title":"Planning, Geometry, and Complexity of Robot Motion","year":"1987","unstructured":"Schwartz, J.T., Sharir, M., Hopcroft, J. (eds.): Planning, Geometry, and Complexity of Robot Motion. Ablex Series in Artificial Intelligence. Ablex Publishing Corporation, Norwood, NJ (1987)"},{"key":"1952_CR28","doi-asserted-by":"crossref","unstructured":"Sommese, A.J., Wampler, II, C.W.: The Numerical Solution of Systems of Polynomials Arising in Engineering and Science. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, (2005)","DOI":"10.1142\/9789812567727"},{"issue":"2","key":"1952_CR29","doi-asserted-by":"publisher","first-page":"245","DOI":"10.1007\/s10957-006-9081-0","volume":"132","author":"Y Yang","year":"2007","unstructured":"Yang, Y.: Globally convergent optimization algorithms on Riemannian manifolds: uniform framework for unconstrained and constrained optimization. J. Optim. Theory Appl. 132(2), 245\u2013265 (2007)","journal-title":"J. Optim. Theory Appl."}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-024-01952-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11075-024-01952-3\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-024-01952-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,9,5]],"date-time":"2025-09-05T23:31:12Z","timestamp":1757115072000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11075-024-01952-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,10,5]]},"references-count":29,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2025,9]]}},"alternative-id":["1952"],"URL":"https:\/\/doi.org\/10.1007\/s11075-024-01952-3","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"value":"1017-1398","type":"print"},{"value":"1572-9265","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,10,5]]},"assertion":[{"value":"28 May 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"29 September 2024","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 October 2024","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare no competing interests.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interest"}}]}}