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Appl. Algebra Geometry"],"published-print":{"date-parts":[[2025,9,30]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>One of the most animated themes of multidimensional persistence is the comparison of invariants. The matching distance between persistent Betti numbers functions (or rank invariants) is among the most studied metrics in this context, particularly in 2-parameter persistence. The main reason for this interest is that, in the 2-parameter case, the foliation method allows for an effective computation of the matching distance, based on filtering the space along lines of positive slope. Our work provides a qualitative analysis, based on a construction called extended Pareto grid, of the filtering lines that contribute to the computation of the matching distance. Under certain genericity assumptions, we show that these lines must either have slope 1 or belong to a collection of special lines associated with discontinuity phenomena.<\/jats:p>","DOI":"10.1137\/24m1680398","type":"journal-article","created":{"date-parts":[[2025,8,6]],"date-time":"2025-08-06T09:00:30Z","timestamp":1754470830000},"page":"554-576","source":"Crossref","is-referenced-by-count":0,"title":["Matching Distance via the Extended Pareto Grid"],"prefix":"10.1137","volume":"9","author":[{"given":"Patrizio","family":"Frosini","sequence":"first","affiliation":[{"name":"University of Pisa, 56127 Pisa, Italy."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0009-9656-6976","authenticated-orcid":true,"given":"Eloy M\u00f3sig","family":"Garc\u00eda","sequence":"additional","affiliation":[{"name":"University of Pisa, 56127 Pisa, Italy."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4494-863X","authenticated-orcid":true,"given":"Nicola","family":"Quercioli","sequence":"additional","affiliation":[{"name":"University of Bologna and WiLab-CNIT, 40136 Bologna, Italy."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Francesca","family":"Tombari","sequence":"additional","affiliation":[{"name":"Max Planck Institute for Mathematics in the Sciences, 04103 Leipzig, Germany."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2025,8,6]]},"reference":[{"key":"ref1","unstructured":"M. Assif P. K. and Y. Baryshnikov, Biparametric Persistence for Smooth Filtrations, preprint, arXiv:2110.09602, 2021."},{"key":"ref2","unstructured":"A. Bapat, R. Brooks, C. Hacker, C. Landi, B. I. Mahler, and E. R. Stephenson, Computing the Matching Distance of 2-Parameter Persistence Modules from Critical Values, preprint, \u00a0https:\/\/arxiv.org\/abs\/2210.12868 (2022)."},{"key":"ref3","doi-asserted-by":"publisher","DOI":"10.1016\/j.patrec.2011.07.014"},{"key":"ref4","doi-asserted-by":"publisher","DOI":"10.1007\/s10851-008-0096-z"},{"key":"ref5","doi-asserted-by":"publisher","DOI":"10.2140\/agt.2023.23.2895"},{"key":"ref6","doi-asserted-by":"publisher","DOI":"10.1002\/mma.2704"},{"key":"ref7","doi-asserted-by":"publisher","DOI":"10.1007\/s41468-019-00041-y"},{"key":"ref8","doi-asserted-by":"publisher","DOI":"10.4208\/jcm.1809-m2018-0043"},{"key":"ref9","doi-asserted-by":"publisher","DOI":"10.1007\/s00454-006-1276-5"},{"key":"ref10","doi-asserted-by":"publisher","DOI":"10.1007\/s10440-008-9332-1"},{"key":"ref11","doi-asserted-by":"publisher","DOI":"10.1007\/s41468-023-00128-7"},{"key":"ref12","first-page":"309","volume":"14","author":"Kerber M.","year":"2023","journal-title":"J. Comput. Geom."},{"key":"ref13","series-title":"LIPIcs Leibniz Int. Proc. Inform. 129","first-page":"46","volume-title":"35th International Symposium on Computational Geometry","author":"Kerber M.","year":"2019"},{"key":"ref14","series-title":"LIPIcs Leibniz Int. Proc. Inform. 164","first-page":"53","volume-title":"36th International Symposium on Computational Geometry","author":"Kerber M.","year":"2020"},{"key":"ref15","unstructured":"E. Miller, Homological Algebra of Modules over Posets, preprint, arXiv:2008.00063, 2020."},{"key":"ref16","unstructured":"Sage, SageMath, the Sage Mathematics Software System (Version 10.2) https:\/\/www.sagemath.org, (2023)."},{"key":"ref17","doi-asserted-by":"publisher","DOI":"10.1016\/0040-9383(75)90002-6"}],"container-title":["SIAM Journal on Applied Algebra and Geometry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/24M1680398","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T13:43:04Z","timestamp":1787233384000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/24M1680398"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,8,6]]},"references-count":17,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2025,9,30]]}},"alternative-id":["10.1137\/24M1680398"],"URL":"https:\/\/doi.org\/10.1137\/24m1680398","relation":{},"ISSN":["2470-6566"],"issn-type":[{"value":"2470-6566","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,8,6]]}}}