[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83088-en":3,"doc-seo-83088-105":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},83088,1099514067415,"Rowan","https://ap-avatar.wpscdn.com/avatar/100002539d78ffe74a7?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779092875211072502",8,"Research & Report","Groebner.jl: Fast Gröbner Tracing in Julia","Groebner.jl introduces a Julia implementation of the F4 algorithm that exposes Traverso’s Gröbner tracing via reusable learn and apply interfaces. The approach speeds repeated Gröbner basis computations that arise in multi-modular and evaluation–interpolation methods by amortizing expensive learning steps. It supports SIMD-friendly coefficient representations, including tuples of machine integers, enabling efficient compiled code. The document also demonstrates applicability to structural identifiability of ODE models and polynomial system solving through rational univariate representations.","arXiv :2607 .06372v2 [ cs . SC] 11 Jul 2026  \nGroebner.jl: Fast Gröbner Tracing in Julia  \nAlexander Demin⋆  \nLaboratoire 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)  \nAbstract. A standard way to control expression swell in computer algebra is to use multi-modular or evaluation-interpolation methods. In computations involving Gröbner bases, these techniques typically require repeatedly computing Gröbner bases of specializations of the same ideal. These repeated computations can be accelerated through precomputation, notably using Traverso’s tracing.  \nWe present Groebner.jl, a Julia implementation of the F4 algorithm that exposes Traverso’s tracing through a reusable public interface. The implementation supports SIMD-friendly coefficient types, such as tuples of machine integers, which Julia compiles to efficient code with little manual intervention. This lets other Julia software leverage tracing to obtain speedups in applications such as structural identifiability of ordinary differential equation models and polynomial system solving.  \nKeywords: Gröbner basis, F4 algorithm, Modular integer arithmetic, Julia, SIMD  \n1 Introduction  \nWhen using multi-modular and evaluation-interpolation methods, it is typical to repeatedly compute Gröbner bases of specializations ofthe same ideal. Traverso’stracing [19] is a method of precomputation that accelerates such repeated calculations. One first performs a learn computation, a modified Gröbner basis computation that additionally records which S-polynomials reduce to zero. One may then perform several apply computations on specializations of the same ideal, omitting reductions of S-polynomials that are expected to reduce to zero.  \nHigh-performance implementations of the F4 algorithm [8] use tracing internally for computing Gröbner bases over the rationals, including msolve [2,15]  \n⋆ This work has been supported by an ERC-2023-ADG grant for the ODELIX project (number 101142171) .  \nFunded by the European Union. Views and opinions expressed  \nare 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  \nand Giac/Xcas [17]; related ideas appear in [13] . Typically, these systems do not expose tracing through their user interfaces. Many applications, however, use Gröbner basis computations as intermediate subroutines rather than final outputs. Therefore, providing a reusable learn & apply interface would make tracing available to a broader class of symbolic computation workflows.  \nWe discuss Groebner.jl, a Julia package that implements the F4 algorithm and exposes Traverso’s tracing through public learn and apply functions [10] . To further amortize computational costs, the apply routine can compute Gröbner bases in batches. The implementation uses SIMD-friendly coefficient types and relies on Julia’s LLVM-based compiler to produce efficient code across architectures with little additional implementation effort. We evaluate these techniques on applications in structural identifiability [5,7] and polynomial system solving via rational univariate representations [6] .  \nAcknowledgements. We would like to thank Roman Pearce, Gleb Pogudin, Fabrice Rouillier, and Chee Yap for useful discussions. We are also grateful to the anonymous referees for helpful comments and suggestions. The project is opensource, and we are grateful to all contributors; in particular, to Matthias Franz.  \n2 Gröbner tracing  \nThis section briefly describes the ideas behind Gröbner tracing using the F4 algorithm as an example. We refer to [19] for details. Tracing consists of two stages, both of which may be viewed as modifications of the classic F4 algorithm, as illustrated in Figure 1:  \nFig. 1.","cbCaiedTQ8a34xax","https://ap.wps.com/l/cbCaiedTQ8a34xax","pdf",650396,3,1,10,"English","en",105,"# Introduction\n## Gröbner tracing with F4\n# Interface design\n## Generic coefficient arithmetic\n## Learn stage\n## Apply stage","[{\"question\":\"What problem does Gröbner tracing target in Gröbner basis computations?\",\"answer\":\"It accelerates repeated Gröbner basis computations on specializations of the same ideal by avoiding redundant work, since many runs share monomial structure but differ in coefficients.\"},{\"question\":\"How do the learn and apply stages work in Traverso-style tracing?\",\"answer\":\"The learn stage performs a modified F4 computation that records which rows reduce to zero, storing trace information independently of coefficients. The apply stage uses that trace to build only the useful Macaulay matrix rows for later specializations, reducing monomial operations.\"},{\"question\":\"What performance features does Groebner.jl provide for coefficient handling?\",\"answer\":\"Groebner.jl supports SIMD-friendly coefficient types, such as tuples of machine integers, relying on Julia’s LLVM-based compilation to generate efficient code with minimal manual intervention.\"}]",1784185114,25,{"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},"groebnerjl-fast-grobner-tracing-in-julia","",{"@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/groebnerjl-fast-grobner-tracing-in-julia/83088/",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","2026-07-24","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 Gröbner tracing target in Gröbner basis computations?","Question",{"text":75,"@type":76},"It accelerates repeated Gröbner basis computations on specializations of the same ideal by avoiding redundant work, since many runs share monomial structure but differ in coefficients.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How do the learn and apply stages work in Traverso-style tracing?",{"text":80,"@type":76},"The learn stage performs a modified F4 computation that records which rows reduce to zero, storing trace information independently of coefficients. The apply stage uses that trace to build only the useful Macaulay matrix rows for later specializations, reducing monomial operations.",{"name":82,"@type":73,"acceptedAnswer":83},"What performance features does Groebner.jl provide for coefficient handling?",{"text":84,"@type":76},"Groebner.jl supports SIMD-friendly coefficient types, such as tuples of machine integers, relying on Julia’s LLVM-based compilation to generate efficient code with minimal manual intervention.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,134],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":22,"doc_module":4,"doc_module_name":46,"category_name":132,"show_sort_weight":22,"slug":133},"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":46,"category_name":136,"show_sort_weight":106,"slug":137},19,"General","general"]