[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84640-en":3,"doc-seo-84640-105":29,"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":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},84640,3848291630094,"Emma Wilson","https://eur-avatar.wpscdn.com/davatar_085a072bc5b1113ac321206ff7593b45",8,"Research & Report","Local Polynomial Factorisation Improving the Montes Algorithm","We present a significant improvement of the Nart–Montes algorithm for factoring polynomials over a complete discrete valuation ring A. The work extends Hensel’s lemma using generalized Newton polygons, yielding a new divide-and-conquer strategy. Under residual characteristic zero or sufficiently large characteristic, approximate roots become convenient type representatives, enabling near-optimal complexity for irreducibility and factorisation, including the extra cost of factorisations above the residue field. For OM-factorisation of F in A[x], the discriminant-valuation dependence is improved by a factor δ relative to prior bounds.","arXiv :2607 .02 153v 1 [ cs . SC] 2 Jul 2026  \nLocal polynomial factorisation: improving the Montes algorithm  \nAdrien Poteaux 1 and Martin Weimann2  \n1 Univ. Lille, CNRS, Centrale Lille, UMR 9189 CRIStAL, F-59000 Lille, France  \n2 LMNO, Universit´e de Caen-Normandie  \nAbstract  \nWe improve significantly the Nart-Montes algorithm for factoring polynomials over a complete discrete valuation ring A. Our first contribution is to extend the Hensel lemma in the context of generalised Newton polygons, from which we derive a new divide and conquer strategy. Also, if A has residual characteristic zero or high enough, we prove that approximate roots are convenient representatives of types, leading finally to an almost optimal complexity both for irreducibility and factorisation issues, plus the cost of factorisations above the residue field. For instance, to compute an OM-factorisation of F ∈ A[x], we improve the complexity results of [3] by a factor δ, the discriminant valuation of F.  \n1 Introduction  \nLet A be a complete discrete valuation ring with residue field F and consider F ∈ A[x], monic and separable of degree d. The aim of this paper is to improve complexity bounds for the factorisation of F. Such a polynomial factorisation is a fundamental task of computer algebra with various applications in number theory and algebraic geometry. As such, our complexity results allow to fasten various computational problems, such as Okutsu frames, integral basis or genus of plane curves (see Section 6 for further details) . Our work is based on the seminal Montes algorithm [10], for which the best known complexity is given in [3] . In [8], the authors conclude their paper by:  \nProbably, an optimal local factorisation algorithm would consist in the application of the Montes algorithm as a fast method to get an Okutsu approximation to each irreducible factor, combined with an efficient “Hensel lift”  \nroutine able to improve these initial approximations by doubling the precision at each iteration. One may speculate that Newton polygons of higher order might also be used to design a similar acceleration procedure.  \nWith S. Pauli, Guardia and Nart answered partially to this question thanks to the singlefactor lifting algorithm [11], that can be viewed as a Newton-like method to lift a single factor with a quadratic convergence. This led to the overall complexity analysis of [3] . In this paper, we answer more precisely to this question, by showing that the classical Hensel algorithm can be adapted to the context of Newton polygon of higher order. We also provide a new divide and conquer strategy using this adapted Hensel algorithm, enabling us to lift all factors of F at the same time, with a complexity almost linear in the size of the output. These two elements allow us to gain a factor d in comparison to the complexity result of [3] . Moreover, following [27], we show that when char(F) ∤ d, we can use approximate roots as strongly optimal representatives of a type 1 . This induces an irreducibility test with a complexity almost linear in δ the valuation of the discriminant of F ; see Theorem 2. This improvement propagates for factorisation with a slightly greater assumption  \nAssumption 1 . char(F) = 0 or char(F) > d  \nleading a complexity almost linear in d n for a required precision n ≥ δ ; see Theorem 4.  \nRelated work. Classical implemented algorithms for factoring polynomials over Qp (see e.g. [4, 6 , 24 , 25]) are based on the Zassenhaus Round Four algorithm, suffering from loss of precision in computing characteristic polynomials. In [11], the authors introduced anew technique as a combination of the Montes algorithm [9, 10] which exploits the Newton polygons of higher order (as initiated in [25]), and a Newton-like single factor lifting. Further complexity improvements are obtained in [3] . The present work is in the same vein, with the notable difference that we introduce a multi factor lifting, which is used in course of the Monte","cbCaimBHjWepJ2rg","https://ap.wps.com/l/cbCaimBHjWepJ2rg","pdf",492076,1,20,"English","en",105,"# Introduction\n# Related work\n# Organisation of the paper\n# Complexity model","[{\"question\":\"What is the main contribution to the Montes-style approach in this work?\",\"answer\":\"The paper improves the Nart–Montes algorithm by extending Hensel’s lemma to the setting of generalized Newton polygons and deriving a new divide-and-conquer strategy for lifting factors.\"},{\"question\":\"Under what conditions are approximate roots useful for representatives of types?\",\"answer\":\"When the residual characteristic of A is zero or high enough, approximate roots provide convenient representatives of types, which then drive near-optimal complexity results for irreducibility and factorisation.\"},{\"question\":\"How does the new method compare to prior complexity bounds for factorisation in A[x]?\",\"answer\":\"For OM-factorisation of F in A[x], the complexity results of earlier work are improved by a factor δ, where δ is the discriminant valuation of F; additionally, the new multi-factor lifting enables an almost linear-in-output-size divide-and-conquer improvement.\"}]",1784197406,50,{"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":86,"head_meta":88,"extra_data":90,"updated_unix":27},"local-polynomial-factorisation-improving-the-montes-algorithm","",{"@graph":35,"@context":85},[36,53,68],{"@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/local-polynomial-factorisation-improving-the-montes-algorithm/84640/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","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 is the main contribution to the Montes-style approach in this work?","Question",{"text":75,"@type":76},"The paper improves the Nart–Montes algorithm by extending Hensel’s lemma to the setting of generalized Newton polygons and deriving a new divide-and-conquer strategy for lifting factors.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Under what conditions are approximate roots useful for representatives of types?",{"text":80,"@type":76},"When the residual characteristic of A is zero or high enough, approximate roots provide convenient representatives of types, which then drive near-optimal complexity results for irreducibility and factorisation.",{"name":82,"@type":73,"acceptedAnswer":83},"How does the new method compare to prior complexity bounds for factorisation in A[x]?",{"text":84,"@type":76},"For OM-factorisation of F in A[x], the complexity results of earlier work are improved by a factor δ, where δ is the discriminant valuation of F; additionally, the new multi-factor lifting enables an almost linear-in-output-size divide-and-conquer improvement.","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":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,114,119,122,126,129,133],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":45,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":45,"category_name":112,"show_sort_weight":28,"slug":113},6,"Technology","technology",{"id":115,"doc_module":4,"doc_module_name":45,"category_name":116,"show_sort_weight":117,"slug":118},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":120,"slug":121},30,"research-report",{"id":123,"doc_module":4,"doc_module_name":45,"category_name":124,"show_sort_weight":21,"slug":125},9,"Religion & Spirituality","religion-spirituality",{"id":21,"doc_module":4,"doc_module_name":45,"category_name":127,"show_sort_weight":21,"slug":128},"World Cup","world-cup",{"id":130,"doc_module":4,"doc_module_name":45,"category_name":131,"show_sort_weight":130,"slug":132},10,"Lifestyle","lifestyle",{"id":134,"doc_module":4,"doc_module_name":45,"category_name":135,"show_sort_weight":106,"slug":136},19,"General","general"]