[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84910-en":3,"doc-seo-84910-105":29,"detail-sidebar-cat-0-en-105":90},{"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":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},84910,1099514068035,"Ezra","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","Fast Rational Univariate Representation via Gaussian Elimination","The note introduces RationalUnivariateRepresentation.jl, a Julia package designed to compute rational univariate representations for zero-dimensional polynomial systems. It focuses on an FGLM-like stage implemented with dense linear algebra and Gaussian elimination, arguing for this practical choice through detailed software implementation. The work demonstrates that the approach can produce guaranteedly correct ideal parametrizations with thousands of solutions within seconds, while aligning performance with contemporary state-of-the-art tools.","arXiv :2607 .06397v2 [ cs . SC] 11 Jul 2026  \nFast Rational Univariate Representation via Gaussian Elimination  \nAlexander Demin 1 ⋆ and Fabrice Rouillier2  \n1 Laboratoire d’informatique de l’École polytechnique,  \nLIX, UMR 7161, CNRS,  \n1 rue Honoré d’Estienne d’Orves,  \n91120 Palaiseau, France  \n[demin@lix.polytechnique.fr](demin@lix.polytechnique.fr)  \n2 Sorbonne Université, CNRS, Inria, Paris, France  \n[Fabrice.Rouillier@inria.fr](Fabrice.Rouillier@inria.fr)  \nAbstract. In this note, we present RationalUnivariateRepresentation.jl, a Julia package for computing rational univariate representations of zerodimensional polynomial systems. The package uses dense linear algebra and Gaussian elimination for the FGLM-like stage. The purpose of this contribution is to advocate for this choice and explain the implementation details that turn the algorithm into practical software. In particular, we show that our implementation can compute guaranteedly correct parametrizations of ideals with thousands of solutions within seconds.  \nKeywords: zero-dimensional polynomial systems, rational univariate representation, FGLM, Gröbner bases, Julia  \n1 Introduction  \nLet k be a field and ¯k its algebraic closure. Let I ⊆ k [x1 , ... , xn] be a zerodimensional ideal. A rational univariate representation [8] of the roots of I in ¯kn consists of a linear form t = a 1 x 1 + ··· + an xn that is injective on the roots, where a 1 , . . . , an ∈ k, together with polynomials f, f1 ,..., fn ∈ k [T] . Each root of I can be recovered from a root β ∈ ¯k of f via xi = fi (β)/f ′(β) . Our goal is to compute such parametrizations of the roots of the radical √I.  \nWhen the ideal is in shape position, a lexicographic Gröbner basis of I yields such a parametrization, and it can be efficiently computed from a degrevlex Gröbner basis using the FGLM algorithm [3] . In general, with a random change of variables, the radical can be placed in shape position with high probability.  \n⋆ Alexander Demin has been supported by an ERC-2023-ADG grant for the ODELIX  \nproject (number 101142171) .  \nFunded by the European Union. Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency. Neither the European Union northe granting authority can be held responsible for them.  \n2 Demin and Rouillier  \nSeveral strategies exploit the sparsity of the multiplication matrices arising in FGLM through the Wiedemann algorithm [9] or Block Krylov methods [4,6]; these are implemented in solvers such as msolve [1] or Giac [7] .  \nIn this paper, we make several practical observations. For example, the multiplication matrices arising in the FGLM algorithm are often only moderately sparse (see Section 5.1), which makes sparse linear algebra less compelling than one might expect. We then discuss how these observations motivate the use of classical Gaussian elimination for computing parametrizations.  \nWe present an optimized deterministic implementation of the algorithm of Demin, Rouillier, and Ruiz [2], based on dense linear algebra and classical Gaussian elimination, and show experimentally that this design leads to competitive performance with state-of-the-art software.  \n2 Background  \n2.1 Shape position  \nLet I ⊆ k [x1 , ... , xn] be a zero-dimensional ideal. Let V(I) ⊆ ¯kn denote the algebraic variety of I. Let D be the dimension of the quotient algebra k[x1 ,..., xn]/I as a k-vector space. Equivalently, D is the number of roots of I counted with multiplicities.  \nA particularly simple parametrization arises when the ideal is in shape position:  \nDefinition 1 (Shape position) . An ideal I is said to be in shape position with respect to the indeterminate xn if the reduced Gröbner basis with respect to the lexicographical ordering with xn \u003C . . . \u003C x 1 is  \nG = {g(xn), x 1 − g1 (xn),..., xn − gn (xn)} , with g, g 1 , . . . , gn ∈ k [T], where deg(g) = D and deg(gi) \u003C D for i ","cbCaiu9zYdaiSZd6","https://ap.wps.com/l/cbCaiu9zYdaiSZd6","pdf",659432,1,10,"English","en",105,"# Introduction\n# Background\n## Shape position\n## From shape position to the general case\n## Verifying the linear form","[{\"question\":\"What problem does the paper address?\",\"answer\":\"It presents a method and software for computing rational univariate representations of the roots of zero-dimensional polynomial ideals, including the radical of the ideal.\"},{\"question\":\"How does the Julia package compute rational univariate representations?\",\"answer\":\"It uses an FGLM-like stage based on dense linear algebra and classical Gaussian elimination, turning the underlying algorithm into practical software.\"},{\"question\":\"Why can a random linear form be used to reach shape position?\",\"answer\":\"With a suitable choice of coefficients over an infinite field, the constructed ideal IT becomes in shape position with respect to the new indeterminate, and the roots of √IT correspond bijectively to those of √I.\"}]",1784199290,25,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":85,"head_meta":87,"extra_data":89,"updated_unix":27},"fast-rational-univariate-representation-via-gaussian-elimination","",{"@graph":35,"@context":84},[36,53,67],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/fast-rational-univariate-representation-via-gaussian-elimination/84910/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":61,"encodingFormat":60,"isAccessibleForFree":62,"interactionStatistic":63},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-16",true,{"@type":64,"interactionType":65,"userInteractionCount":4},"InteractionCounter",{"@type":66},"ViewAction",{"@type":68,"mainEntity":69},"FAQPage",[70,76,80],{"name":71,"@type":72,"acceptedAnswer":73},"What problem does the paper address?","Question",{"text":74,"@type":75},"It presents a method and software for computing rational univariate representations of the roots of zero-dimensional polynomial ideals, including the radical of the ideal.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"How does the Julia package compute rational univariate representations?",{"text":79,"@type":75},"It uses an FGLM-like stage based on dense linear algebra and classical Gaussian elimination, turning the underlying algorithm into practical software.",{"name":81,"@type":72,"acceptedAnswer":82},"Why can a random linear form be used to reach shape position?",{"text":83,"@type":75},"With a suitable choice of coefficients over an infinite field, the constructed ideal IT becomes in shape position with respect to the new indeterminate, and the roots of √IT correspond bijectively to those of 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