[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81849-en":3,"doc-seo-81849-105":31,"detail-sidebar-cat-0-en-105":93},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":28,"seo_description":14,"update_tm":29,"read_time":30},81849,5909877438554,"Maeve","https://ap-avatar.wpscdn.com/avatar/5600025385ad2bf12a7?_k=1778553567797529272",8,"Research & Report","Computing the Continuous Symmetries of a Parametrized Variety","Computing the symmetry Lie algebra of a parametrized algebraic variety can be done directly from its parametrization, avoiding computation of the variety’s vanishing ideal. The work develops practical polynomial-time Monte Carlo algorithms to construct the symmetry Lie algebra. It also establishes methods to test binomiality of the ideal after invertible coordinate changes. Applications include staged tree model varieties, colored Gaussian graphical models, rational curves, and characterization results for symmetries of many secant varieties.","arXiv :2607 .02676v1 [math .AG] 2 Jul 2026  \nCOMPUTING THE CONTINUOUS SYMMETRIES  \nOF A PARAMETRIZED VARIETY  \nBENJAMIN BIAGGI, JAN DRAISMA, FULVIO GESMUNDO, AIDA MARAJ, AND MAGDALÉNA MIŠINOVÁ  \nAbstract . We prove that the symmetry Lie algebra of a parametrized variety can be determined directly from the parametrization, without computing the vanishing ideal of the variety. We derive a practical polynomial-time Monte Carlo algorithm for computing the symmetry Lie algebra of a parametrized variety. We discuss applications to testing the binomiality of the ideal of a parametrized variety after changing coordinates, and test this property on varieties arising from staged tree models and colored Gaussian graphical models.  \nFinally, we discuss symmetries and binomiality after changing coordinates for rational curvesand give a characterization of the symmetries of many secant varieties.  \n1. Introduction  \nIn applications of algebraic geometry, one is often given a polynomial map or, more generally, a rational map φ : Cm 99K Cn, and wants to compute properties of the image closure  \nX := {φ(p) | p ∈ Cm and φ is defined at p} .  \nThe closure and other topological notions throughout the paper refer to the Zariski topology; in this setting, the closure coincides with that in the Euclidean topology. This framework yields a large class of (irreducible) algebraic varieties, called unirational varieties.  \nLet GLn := GLn(C) and gln := gln(C) be the general linear group and its Lie algebra, respectively. An important invariant of the embedded variety X is the group  \nGX := {g ∈ GLn | gX = X}  \nof linear symmetries of X , often called the symmetry group or the linear preserver subgroup of X . Knowledge of GX can be exploited in the computation of other properties of X . For instance, the vanishing ideal of X in the polynomial ring C [y1 ,..., yn] is a GX-module, and thus knowledge of the representation theory of GX may help in determining (interesting subspacesof) that vanishing ideal.  \nBy construction, GX is a Zariski closed subgroup of GLn, and hence the union of finitely many connected components, which by standard algebraic group theory are also irreducible. Let G0X be the connected component containing the identity element Idn ∈ GLn. This is a closed normal subgroup of GX, and GX/G0X is a finite group; see, e.g., [Bor91 , §1.2] . The identity component G0X is uniquely determined by the Lie algebra gX of GX . We call gX the symmetry Lie algebra of X .  \nThe problem of determining the symmetry group of an algebraic variety appears in several contexts [GLŠ00 , LP01], in particular in the setting of varieties of matrices, tensors, or operators with special structure [LT92 , TW03 , Joh11] . The symmetry Lie group of varieties of  \n2020 Mathematics Subject Classification. 15A86, 17B45, 68W30, 14Q20 .  \nKey words and phrases. Symmetry Lie algebra, unirational variety, secant variety, linear preserver problem. BB is funded by JD’s Swiss National Science Foundation project grant 200021-227864 .  \nCOMPUTING THE SYMMETRY ALGEBRA 2  \nmatrices of bounded rank was determined in [GLŠ00], and [Wes67] determined the symmetry group of the variety of rank-one tensors. More recently, [GHL25] generalized this result to a wide range of secant varieties of tensors and other related varieties. In [MP26], the symmetry Lie algebra of an algebraic variety X is computed from the polynomials vanishing on X , and in [KV25 , MP26], gX is used to test whether the vanishing ideal of X is binomial after an invertible linear change of variables.  \nThe purpose of this paper is to show that the Lie algebra gX of GX can be computed directly from the parametrization φ without first computing the defining ideal of X . We give practical polynomial-time Monte Carlo algorithms for computing gX and detecting binomiality after an invertible linear change of variables. We show that this insight is useful to derive theoretical results as well: we apply it to varieties of staged t","cbCaiphYxMThAOZ0","https://ap.wps.com/l/cbCaiphYxMThAOZ0","pdf",878239,5,1,26,"English","en",105,"# Introduction\n## Computing the symmetry Lie algebra\n## Polynomial-time Monte Carlo algorithms\n## Testing binomiality after changing coordinates\n## Applications to models and classical varieties\n## Organization and notation","[{\"question\":\"What does the paper compute about a parametrized variety?\",\"answer\":\"It computes the symmetry Lie algebra of the linear symmetry group of the parametrized variety, deriving it directly from the parametrization.\"},{\"question\":\"How are the symmetry Lie algebra computations performed without using the vanishing ideal?\",\"answer\":\"The symmetry Lie algebra is characterized as linear maps that send each point of the variety into its tangent space, which can be accessed through the Jacobian of the parametrization.\"},{\"question\":\"How is binomiality tested after changing coordinates?\",\"answer\":\"The paper develops polynomial-time Monte Carlo procedures that detect whether the vanishing ideal becomes binomial after an invertible linear change of variables.\"}]","Computing the Continuous Symmetries of a Parametrized Variety | 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does the paper compute about a parametrized variety?","Question",{"text":77,"@type":78},"It computes the symmetry Lie algebra of the linear symmetry group of the parametrized variety, deriving it directly from the parametrization.","Answer",{"name":80,"@type":75,"acceptedAnswer":81},"How are the symmetry Lie algebra computations performed without using the vanishing ideal?",{"text":82,"@type":78},"The symmetry Lie algebra is characterized as linear maps that send each point of the variety into its tangent space, which can be accessed through the Jacobian of the parametrization.",{"name":84,"@type":75,"acceptedAnswer":85},"How is binomiality tested after changing coordinates?",{"text":86,"@type":78},"The paper develops polynomial-time Monte Carlo procedures that detect whether the vanishing ideal becomes binomial after an invertible linear change of 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