{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,19]],"date-time":"2026-03-19T03:40:41Z","timestamp":1773891641346,"version":"3.50.1"},"reference-count":39,"publisher":"American Mathematical Society (AMS)","issue":"357","license":[{"start":{"date-parts":[[2026,1,28]],"date-time":"2026-01-28T00:00:00Z","timestamp":1769558400000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/501100001725","name":"Royal Swedish Academy of Sciences","doi-asserted-by":"publisher","award":["[NT:2022-03303]"],"award-info":[{"award-number":["[NT:2022-03303]"]}],"id":[{"id":"10.13039\/501100001725","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"

We introduce a theory of relative tangency for projective algebraic varieties. The dual variety \n\n \n \n X<\/mml:mi>\n Z<\/mml:mi>\n \u2228<\/mml:mo>\n <\/mml:msubsup>\n X_Z^\\vee<\/mml:annotation>\n <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> of a variety \n\n \n X<\/mml:mi>\n X<\/mml:annotation>\n <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> relative to a subvariety \n\n \n Z<\/mml:mi>\n Z<\/mml:annotation>\n <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> is the set of hyperplanes tangent to \n\n \n X<\/mml:mi>\n X<\/mml:annotation>\n <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> at a point of \n\n \n Z<\/mml:mi>\n Z<\/mml:annotation>\n <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>. We also introduce the concept of polar classes of \n\n \n X<\/mml:mi>\n X<\/mml:annotation>\n <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> relative to \n\n \n Z<\/mml:mi>\n Z<\/mml:annotation>\n <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>. We explore the duality of varieties of low rank matrices relative to special linear sections. In this framework, we study the critical points of the Euclidean Distance (ED) function from a data point to \n\n \n X<\/mml:mi>\n X<\/mml:annotation>\n <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>, lying on \n\n \n Z<\/mml:mi>\n Z<\/mml:annotation>\n <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>. The locus where the number of such conditional critical points is positive is called the ED data locus of \n\n \n X<\/mml:mi>\n X<\/mml:annotation>\n <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> given \n\n \n Z<\/mml:mi>\n Z<\/mml:annotation>\n <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>. The generic number of such critical points defines the conditional ED degree of \n\n \n X<\/mml:mi>\n X<\/mml:annotation>\n <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> given \n\n \n Z<\/mml:mi>\n Z<\/mml:annotation>\n <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>. We show the irreducibility of ED data loci, and we compute their dimensions and degrees in terms of relative characteristic classes.<\/p>","DOI":"10.1090\/mcom\/4047","type":"journal-article","created":{"date-parts":[[2024,11,20]],"date-time":"2024-11-20T10:46:20Z","timestamp":1732099580000},"page":"477-524","source":"Crossref","is-referenced-by-count":2,"title":["Conditional Euclidean distance optimization via relative tangency"],"prefix":"10.1090","volume":"95","author":[{"given":"Sandra","family":"Di Rocco","sequence":"first","affiliation":[]},{"given":"Lukas","family":"Gustafsson","sequence":"additional","affiliation":[]},{"given":"Luca","family":"Sodomaco","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2025,1,28]]},"reference":[{"key":"1","series-title":"Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-5323-3","volume-title":"Geometry of algebraic curves. Vol. I","volume":"267","author":"Arbarello, E.","year":"1985","ISBN":"https:\/\/id.crossref.org\/isbn\/0387909974"},{"issue":"2","key":"2","doi-asserted-by":"publisher","first-page":"535","DOI":"10.4418\/2021.76.2.15","article-title":"The maximum likelihood degree of linear spaces of symmetric matrices","volume":"76","author":"Am\u00e9ndola, C.","year":"2021","journal-title":"Matematiche (Catania)","ISSN":"https:\/\/id.crossref.org\/issn\/0373-3505","issn-type":"print"},{"key":"3","volume-title":"Determinants and Matrices","author":"Atiken, A.C.","year":"1939"},{"key":"4","doi-asserted-by":"crossref","unstructured":"[BBE{\\etalchar{+}}21] P. W. Blair, J. Brandsma, N. J. Epsi, S. A. Richard, D. Striegel, J. Chenoweth, R. Mehta, E. Clemens, A. Malloy, C. Lanteri, J. S. Dumler, D. Tribble, T. Burgess, S. Pollett, B. Agan, and D. Clark, Phenotypic differences between distinct immune biomarker clusters during the \u2018hyperinflammatory\u2019 middle-phase of COVID-19, Open Forum Infect. Dis. 8, supplement 1, S320\u2013S321.","DOI":"10.1093\/ofid\/ofab466.637"},{"key":"5","first-page":"3","article-title":"Semidefinite optimization","author":"Parrilo, Pablo A.","year":"2013"},{"issue":"3","key":"6","doi-asserted-by":"crossref","first-page":"671","DOI":"10.1353\/ajm.2006.0019","article-title":"The maximum likelihood degree","volume":"128","author":"Catanese, Fabrizio","year":"2006","journal-title":"Amer. J. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9327","issn-type":"print"},{"key":"7","doi-asserted-by":"publisher","first-page":"855","DOI":"10.1016\/j.jsc.2020.10.005","article-title":"Wasserstein distance to independence models","volume":"104","author":"\u00c7elik, T\u00fcrk\u00fc \u00d6zl\u00fcm","year":"2021","journal-title":"J. Symbolic Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0747-7171","issn-type":"print"},{"key":"8","doi-asserted-by":"publisher","first-page":"351","DOI":"10.1016\/j.jsc.2020.07.009","article-title":"Voronoi cells of varieties","volume":"109","author":"Cifuentes, Diego","year":"2022","journal-title":"J. Symbolic Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0747-7171","issn-type":"print"},{"key":"9","unstructured":"[Cul13] C. E. Cullis, Matrices and Determinoids: Volume 1, Calcutta University Readership Lectures, Cambridge University Press, 1913."},{"key":"10","doi-asserted-by":"crossref","unstructured":"[DHMN24] G. DePaul, S. Ho\u015ften, N. Metya, and I. Nometa, Degrees of the Wasserstein distance to small toric models, arXiv:2402.09626, 2024.","DOI":"10.2140\/astat.2024.15.249"},{"issue":"1","key":"11","doi-asserted-by":"publisher","first-page":"99","DOI":"10.1007\/s10208-014-9240-x","article-title":"The Euclidean distance degree of an algebraic variety","volume":"16","author":"Draisma, Jan","year":"2016","journal-title":"Found. Comput. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1615-3375","issn-type":"print"},{"issue":"1","key":"12","doi-asserted-by":"publisher","first-page":"227","DOI":"10.1137\/19M1265776","article-title":"The bottleneck degree of algebraic varieties","volume":"4","author":"Di Rocco, Sandra","year":"2020","journal-title":"SIAM J. Appl. Algebra Geom."},{"issue":"6","key":"13","doi-asserted-by":"publisher","first-page":"1209","DOI":"10.1007\/s10208-014-9194-z","article-title":"The number of singular vector tuples and uniqueness of best rank-one approximation of tensors","volume":"14","author":"Friedland, Shmuel","year":"2014","journal-title":"Found. Comput. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1615-3375","issn-type":"print"},{"key":"14","series-title":"Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-1700-8","volume-title":"Intersection theory","volume":"2","author":"Fulton, William","year":"1998","ISBN":"https:\/\/id.crossref.org\/isbn\/354062046X","edition":"2"},{"key":"15","series-title":"Pure and Applied Mathematics","isbn-type":"print","volume-title":"Principles of algebraic geometry","author":"Griffiths, Phillip","year":"1978","ISBN":"https:\/\/id.crossref.org\/isbn\/0471327921"},{"key":"16","series-title":"Mathematics: Theory \\& Applications","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-8176-4771-1","volume-title":"Discriminants, resultants, and multidimensional determinants","author":"Gel\u2032fand, I. M.","year":"1994","ISBN":"https:\/\/id.crossref.org\/isbn\/0817636609"},{"key":"17","unstructured":"[GS97] D. Grayson and M. Stillman, Macaulay 2\u2013a system for computation in algebraic geometry and commutative algebra, 1997, available at \\url{http:\/\/www2.macaulay2.com}."},{"key":"18","series-title":"Graduate Texts in Mathematics, No. 52","isbn-type":"print","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4757-3849-0","volume-title":"Algebraic geometry","author":"Hartshorne, Robin","year":"1977","ISBN":"https:\/\/id.crossref.org\/isbn\/0387902449"},{"key":"19","series-title":"Graduate Texts in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-2189-8","volume-title":"Algebraic geometry","volume":"133","author":"Harris, Joe","year":"1992","ISBN":"https:\/\/id.crossref.org\/isbn\/0387977163"},{"issue":"2","key":"20","doi-asserted-by":"publisher","first-page":"145","DOI":"10.1007\/BF01259325","article-title":"The geometric and numerical properties of duality in projective algebraic geometry","volume":"61","author":"Holme, Audun","year":"1988","journal-title":"Manuscripta Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-2611","issn-type":"print"},{"issue":"3","key":"21","doi-asserted-by":"publisher","first-page":"1177","DOI":"10.1080\/00927872.2016.1172632","article-title":"The data singular and the data isotropic loci for affine cones","volume":"45","author":"Horobe\u0163, Emil","year":"2017","journal-title":"Comm. Algebra","ISSN":"https:\/\/id.crossref.org\/issn\/0092-7872","issn-type":"print"},{"issue":"12","key":"22","doi-asserted-by":"publisher","first-page":"Paper No. 107144, 15","DOI":"10.1016\/j.jpaa.2022.107144","article-title":"Data loci in algebraic optimization","volume":"226","author":"Horobe\u0163, Emil","year":"2022","journal-title":"J. Pure Appl. Algebra","ISSN":"https:\/\/id.crossref.org\/issn\/0022-4049","issn-type":"print"},{"key":"23","unstructured":"[HT23] E. Horobe\u0163 and E. T. Turatti, When does subtracting a rank-one approximation decrease tensor rank?, arXiv:2303.14985, 2023."},{"key":"24","unstructured":"[KKS21] K. Kubjas, O. Kuznetsova, and L. Sodomaco, Algebraic degree of optimization over a variety with an application to \ud835\udc5d-norm distance degree, Acta Univ. Sapientiae Math., Special issue on metric algebra geometry, To appear, arXiv:2105.07785, 2021."},{"key":"25","first-page":"163","article-title":"Tangency and duality","author":"Kleiman, Steven L.","year":"1986"},{"key":"26","unstructured":"[KMQS23] K. Kozhasov, A. Muniz, Y. Qi, and L. Sodomaco, On the minimal algebraic complexity of the rank-one approximation problem for general inner products, arXiv:2309.15105, 2023."},{"key":"27","series-title":"Graduate Studies in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1090\/gsm\/128","volume-title":"Tensors: geometry and applications","volume":"128","author":"Landsberg, J. M.","year":"2012","ISBN":"https:\/\/id.crossref.org\/isbn\/9780821869079"},{"key":"28","doi-asserted-by":"crossref","unstructured":"[MRW24] C. Meroni, B. Reinke, and K. Wang, The algebraic degree of the Wasserstein distance, arXiv:2401.12735, 2024.","DOI":"10.1007\/s44426-025-00004-0"},{"key":"29","doi-asserted-by":"crossref","unstructured":"[NLC11] M. Nicolau, A. J. Levine, and G. Carlsson, Topology based data analysis identifies a subgroup of breast cancers with a unique mutational profile and excellent survival, Proc. Nat. Acad. Sci. 108 (2011), no. 17, 7265\u20137270.","DOI":"10.1073\/pnas.1102826108"},{"issue":"1","key":"30","doi-asserted-by":"publisher","first-page":"485","DOI":"10.1137\/080716670","article-title":"Algebraic degree of polynomial optimization","volume":"20","author":"Nie, Jiawang","year":"2009","journal-title":"SIAM J. Optim.","ISSN":"https:\/\/id.crossref.org\/issn\/1052-6234","issn-type":"print"},{"issue":"4","key":"31","doi-asserted-by":"publisher","first-page":"1219","DOI":"10.1090\/S0002-9939-2012-11404-2","article-title":"Matrices with eigenvectors in a given subspace","volume":"141","author":"Ottaviani, Giorgio","year":"2013","journal-title":"Proc. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9939","issn-type":"print"},{"issue":"1","key":"32","doi-asserted-by":"publisher","first-page":"73","DOI":"10.1007\/s12215-021-00600-2","article-title":"Tensors with eigenvectors in a given subspace","volume":"71","author":"Ottaviani, Giorgio","year":"2022","journal-title":"Rend. Circ. Mat. Palermo (2)","ISSN":"https:\/\/id.crossref.org\/issn\/0009-725X","issn-type":"print"},{"key":"33","first-page":"109","article-title":"Compound matrices and Pfaffians: a representation of geometric algebra","author":"Prells, Uwe","year":"2002"},{"issue":"2","key":"34","doi-asserted-by":"crossref","first-page":"247","DOI":"10.24033\/asens.1346","article-title":"Polar classes of singular varieties","volume":"11","author":"Piene, Ragni","year":"1978","journal-title":"Ann. Sci. \\'{E}cole Norm. Sup. (4)","ISSN":"https:\/\/id.crossref.org\/issn\/0012-9593","issn-type":"print"},{"key":"35","doi-asserted-by":"publisher","first-page":"139","DOI":"10.1007\/978-3-319-15081-9_8","article-title":"Polar varieties revisited","author":"Piene, Ragni","year":"2015"},{"key":"36","first-page":"203","article-title":"Dualities","author":"Rostalski, Philipp","year":"2013"},{"key":"37","doi-asserted-by":"publisher","first-page":"74","DOI":"10.1016\/j.jsc.2022.04.016","article-title":"Degrees of Kalman varieties of tensors","volume":"114","author":"Shahidi, Zahra","year":"2023","journal-title":"J. Symbolic Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0747-7171","issn-type":"print"},{"issue":"4","key":"38","doi-asserted-by":"publisher","first-page":"603","DOI":"10.1007\/s10463-010-0295-4","article-title":"Multivariate Gaussian, semidefinite matrix completion, and convex algebraic geometry","volume":"62","author":"Sturmfels, Bernd","year":"2010","journal-title":"Ann. Inst. Statist. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0020-3157","issn-type":"print"},{"key":"39","unstructured":"[{The}22] The Sage Developers, SageMath, the Sage Mathematics software system (version 9.5), 2022, \\url{https:\/\/www.sagemath.org}."}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.ams.org\/mcom\/2026-95-357\/S0025-5718-2025-04047-X\/mcom4047_AM.pdf","content-type":"application\/pdf","content-version":"am","intended-application":"syndication"},{"URL":"https:\/\/www.ams.org\/mcom\/2026-95-357\/S0025-5718-2025-04047-X\/S0025-5718-2025-04047-X.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,9,16]],"date-time":"2025-09-16T17:51:15Z","timestamp":1758045075000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2026-95-357\/S0025-5718-2025-04047-X\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,1,28]]},"references-count":39,"journal-issue":{"issue":"357","published-print":{"date-parts":[[2026,1]]}},"alternative-id":["S0025-5718-2025-04047-X"],"URL":"https:\/\/doi.org\/10.1090\/mcom\/4047","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2025,1,28]]}}}