[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82226-en":3,"doc-seo-82226-105":30,"detail-sidebar-cat-0-en-105":91},{"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":13,"seo_description":14,"update_tm":28,"read_time":29},82226,1374391974468,"Eden","https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0",8,"Research & Report","Beyond F5 and GVW Proper Cover Algorithm for Fast Ideal Basis Computation","Gröbner basis computation is expensive, particularly under lexicographic order. F5 and the GVW variant are key efficient solvers, yet the proper basis algorithm offers a parameterized ideal framework without modern signature-based optimizations. This work introduces the ProperCover algorithm for zero-dimensional polynomial ideals by merging GVW cover optimization over signatures with proper basis theory. It generalizes signature, cover, POT ordering, reduction, and S-pairs to parameterized coefficients, builds a two-phase compatible-factor method with hungry refinement, and proves termination and correctness, improving performance versus F5 and GVW.","Beyond F5 and GVW: The Proper-Cover Algorithm for Fast Ideal Basis Computation  \nSheng-Ming Ma BeiHang University [smmath@foxmail. com](smmath@foxmail. com)[masm@pku. org. cn](masm@pku. org. cn)  \nYi Liu BeiHang University [liuyee@buaa. edu. cn](liuyee@buaa. edu. cn)  \narXiv :2607 .09 163v 1 [ cs . SC] 10 Jul 2026  \nZheng-Lin Jiao  \nBeiHang University  \n[regulusjiao@gmail. com](regulusjiao@gmail. com)  \nAbstract  \nGröbner basis computation incurs heavy computational overhead, especially under lexicographic order. F5 and its GVW variant dominate efficient field-based Gröbner basis solving. The proper basis algorithm offers a parameterized ideal computation framework without leveraging modern signature-based optimizations. This work presents the ProperCover algorithm for zero-dimensional polynomial ideals by combining GVW’s cover optimization over signature with the proper basis theory. We generalize signature, cover, POT ordering, reduction and S-pair concepts to parameterized coefficients, design a twophase algorithm with compatible factor construction and hungry refinement, and rigorously prove termination and output correctness. Accordingly, we propose a new framework for the efficient computation of polynomial ideal bases. Benchmark results show that Proper-Cover surpasses F5 under all monomial orderings and delivers clear speedups over GVW for lexicographic (plex) order.  \n1 Introduction  \nSince Buchberger introduced his celebrated algorithm in his seminal PhD thesis, the theory of Gröbner bases has become a standard tool in computer algebra. However, the computation of Gröbner bases is often plagued by high complexity, especially with respect to the lexicographic order. Up to the present, Faugère’s F5 algorithm[4, 6] and its extended variants remain the most efficient approaches for computing Gröbner bases in practice. The GVW algorithm[14] revisesand optimizes the cover mechanism inherent to F5, yielding drastically enhanced computational efficiency.  \nWhile Gröbner bases over a field have been generalized to settings over rings, especially principal ideal rings[1, 10, 17], these ring-based generalizations have not yet been applied to improve the computation of polynomial ideals ov˜ er a field. The proper basis algorithm[15] defines and computes an ideal basis over variables x =˜ (x 2 ,..., xn) with the least variable x 1 acts  \nas a parameter within the polynomial algebra K [x 1][x] . It outperforms Faugère’s F4 algorithm, Buchberger’s classical algorithm and Möller’s algorithm.  \nTheorem 1.1 (Main result). For every zero-dimensional input ideal I = ⊂ K[x 1][x˜xx], the Proper  \nCover algorithm terminates and outputs pairs (q j , Bq j ) such that the eliminant χ = ∏ jq j and ⟨LT(πq j (I))⟩ = ⟨LT(Bq j )⟩ in (K[x 1]/ (q j))[x˜xx] for every j.  \nConsequently, B := Sj(Bq j ∪{q j}) is aproper basis of I.  \nIn this paper we combine the GVW algorithm and the proper basis algorithm to develop the Proper-Cover algorithm. It delivers better computational performance than Faugère’s F5 algorithm for all monomial orderings, and clearly outperforms the GVW algorithm under lexicographic (plex) ordering.  \nIn Section 2, we extend the notions of signature, cover and POT monomial ordering from the setting of base fields to parameterized coefficients over a principal ideal domain (PID) . Within this section, we formalize the definitions of proper basis and regular reduction, and introduce S-pairs and semi-S-pairs as extensions of classical S-polynomials.  \nIn Section 3, we introduce our two-phase algorithm consisting of Algorithm 1 and Algorithm 2. Relying on this two-phase algorithm, we further introduce the notion of compatible factors.  \nIn Section 4, we establish theoretical soundness of our Proper-Cover algorithm: termination and correctness of its output proper basis. We additionally demonstrate that the compatible factors generated by Algorithm 1, combined with the hungry refinement from Algorithm 2, produce the valid eliminant. The","cbCaiosTRO1EqSSa","https://ap.wps.com/l/cbCaiosTRO1EqSSa","pdf",200107,3,1,15,"English","en",105,"# Introduction\n## Proper basis, signatures, and cover over parameterized coefficients\n# Two-phase Proper-Cover algorithm\n## Termination and correctness\n# Benchmarks and experimental results","[{\"question\":\"What problem does the ProperCover algorithm address compared with F5 and GVW?\",\"answer\":\"It targets the heavy overhead of Gröbner basis computation, especially for lexicographic order, by combining GVW-style cover optimization with the proper basis framework.\"},{\"question\":\"How does ProperCover generalize core GVW/F5 concepts in this paper?\",\"answer\":\"It extends signature, cover, POT ordering, regular reduction, and S-pairs (including semi-S-pairs) to parameterized coefficients over rings/quotients.\"},{\"question\":\"What guarantees does the paper provide about ProperCover’s output?\",\"answer\":\"The paper rigorously proves that the algorithm terminates and outputs a proper basis, matching leading-term/eliminant relationships required for correctness.\"}]",1784178968,38,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"beyond-f5-and-gvw-proper-cover-algorithm-for-fast-ideal-basis-computation","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":20},"https://docshare.wps.com/document/research-report/",{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/beyond-f5-and-gvw-proper-cover-algorithm-for-fast-ideal-basis-computation/82226/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-21","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What problem does the ProperCover algorithm address compared with F5 and GVW?","Question",{"text":75,"@type":76},"It targets the heavy overhead of Gröbner basis computation, especially for lexicographic order, by combining GVW-style cover optimization with the proper basis framework.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does ProperCover generalize core GVW/F5 concepts in this paper?",{"text":80,"@type":76},"It extends signature, cover, POT ordering, regular reduction, and S-pairs (including semi-S-pairs) to parameterized coefficients over rings/quotients.",{"name":82,"@type":73,"acceptedAnswer":83},"What guarantees does the paper provide about ProperCover’s output?",{"text":84,"@type":76},"The paper rigorously proves that the algorithm terminates and outputs a proper basis, matching leading-term/eliminant relationships required for 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