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The recent algorithms for computing them consider the corresponding ideal\n                    <jats:italic>J<\/jats:italic>\n                    in\n                    <jats:italic>K<\/jats:italic>\n                    [\n                    <jats:italic>A<\/jats:italic>\n                    ,\u00a0\n                    <jats:italic>X<\/jats:italic>\n                    ], and are based on stability of Gr\u00f6bner bases of ideals under specializations of the parameters\n                    <jats:italic>A<\/jats:italic>\n                    . Starting from a Gr\u00f6bner basis of\n                    <jats:italic>J<\/jats:italic>\n                    , the computation splits recursively depending on the vanishing of the evaluation of some \u201ccoefficients\u201d in\n                    <jats:italic>K<\/jats:italic>\n                    [\n                    <jats:italic>A<\/jats:italic>\n                    ]. In this paper, taking inspiration from the algorithm described by Nabeshima, we create a new iterative algorithm to compute comprehensive Gr\u00f6bner systems. We show how we keep track of the sub-cases to be considered, and how we avoid some redundant computation branches using \u201ccomparatively-cheap\u201d ideal-membership tests, instead of radical-membership tests.\n                  <\/jats:p>","DOI":"10.1007\/s00200-025-00684-8","type":"journal-article","created":{"date-parts":[[2025,4,1]],"date-time":"2025-04-01T06:20:45Z","timestamp":1743488445000},"page":"681-694","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A new iterative algorithm for comprehensive Gr\u00f6bner systems"],"prefix":"10.1007","volume":"37","author":[{"given":"Anna Maria","family":"Bigatti","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Elisa","family":"Palezzato","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Michele","family":"Torielli","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2025,3,31]]},"reference":[{"key":"684_CR1","unstructured":"Abbott, J., Bigatti, A. 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