[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84094-en":3,"doc-seo-84094-105":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},84094,1099514067438,"River Wang","https://ap-avatar.wpscdn.com/avatar/100002539ee87300030?x-image-process=image/resize,m_fixed,w_180,h_180&k=1780474512215547542",8,"Research & Report","Computing Singular Solutions of Polynomial Systems: Towards Superlinear Convergence Without Deflation","Numerical Algebraic Geometry often computes isolated solutions of polynomial systems by tracking a homotopy curve, but this becomes difficult near singular roots in the endgame regime. The work introduces an Arclength Endgame for corank-1 systems, proving superlinear convergence in a neighborhood while using only evaluations of the system and its Jacobian, avoiding additional derivative computations. For higher corank, it proposes a Lifted Arclength Endgame with encouraging experiments. A new, more stable method estimates Puiseux series exponents beyond the first ratio.","arXiv :2607 .06329v1 [math .NA] 7 Jul 2026  \nComputing singular solutions of polynomial systems: towards superlinear convergence without deflation  \nMikhail Karapetyants Vladimir Kolmogorov Jeferson Zapata  \nInstitute of Science and Technology Austria (ISTA)  \n{mikhail. karapetyants,vnk,[jeferson. zapata](jeferson. zapata}@ist. ac. at)[}](jeferson. zapata}@ist. ac. at)[@ist. ac. at](jeferson. zapata}@ist. ac. at)  \nAbstract  \nIn Numerical Algebraic Geometry (NAG) isolated solutions of polynomial systems are usually computed by tracking a solution curve defined by a homotopy equation. The tracking problem becomes especially challenging close to a singular root (the “endgame” regime) . Existing approaches include power series endgames, Cauchy endgames, and various methods that regularize the system via dual-space-based deflation. We make the following contributions.  \n(1) For corank-1 systems we introduce a new “Arclength Endgame” which combines the idea of the classical pseudo-arclength continuation method with the estimation of the Puiseux series of the curve. We formally prove that it has a superlinear rate of convergence in some neighborhood of the root. The method uses only evaluations of the system and its Jacobian, whereas previous techniques with proven superlinear convergence (such as deflation) require computing additional derivatives of the system.  \n(2) For systems with a larger corank we propose a heuristic “Lifted Arclength Endgame”, which shows promising experimental results.  \n(3) A key step in our approach (as well as in the standard power series endgame) is esti  \nmating the Puiseux series of the curve, which is characterized by fractional exponents ki /c for i ≥ 1 together with associated coefficients. Previous work addressed only estimating the ratio k 1 /c. We present a new method for that which empirically appears to be more stable than previous methods, and also show how to estimate ki /c for i ≥ 2.  \n1 Introduction  \nWe consider the problem of numerically solving a system of equations f (z) = 0 with a zerodimensional set of solutions. Here f is an analytic mapping Cn → Cn. A standard approach in Numerical Algebraic Geometry (NAG) for tackling this problem is as follows. First, one constructsa homotopy function h (z, t) = (1 − t)f (z) + tg(z) where g is a polynomial system with easily computable roots. Consider one such root zroot. If g is chosen generically then there exists a unique smooth function z : (0 , 1] → Cn with z(1) = zroot and h (z(t), t) = 0 for all t ∈ (0 , 1] . Furthermore, if z((0, 1]) is bounded then the limit z ∗ = z(0) = limt→0 z (t) exists and is a root off. The latter condition will always be satisfied if system f is homogeneous. If g is chosen to have sufficiently many roots then every root of f will be covered with probability 1 [SW05, Theorem 8.4.1] .  \nBy differentiating equation h (z(t), t) = 0 with respect to t one obtains Davidenko ODE:  \nz˙(t) = −hz (z(t), t)−1ht(z(t), t) (1)  \nWe now need to track curve z (t) by numerically solving this ODE.  \nThis problem becomes especially challenging when t approaches zero and z ∗ = z(0) is a singular solution z ∗ , i.e. the Jacobian fz(z∗ ) is singular. This regime, referred to as the “endgame”, constitutes the main focus of the present work. The following assumptions are maintained throughout the manuscript:  \nAssumption 1 . (a) h : Cn+1 → Cn is a polynomial mapping, and f (z) = h (z,0) .  \n(b) Point z ∗ ∈ Cn is an isolated solution of f (z) . We denote x∗ = (z∗ , 0) , J ∗ = hz(x∗ ) = fz(z∗ ) and h∗t = ht(x∗ ) . We also let κ = n − rank(J∗ ) be the corank of J ∗ .  \n(c) rank (J∗ ) \u003C n, i. e. z ∗ is a singular solution of f.  \n(d) h∗t is linearly independent of columns in J ∗ , i. e.  \nrank([J∗ |h∗t]) = rank(J∗ ) + 1 .  \n(e) There exists a finite set Π of formal Puiseux series of the form  \n∞  \nz (t) = z∗ +X ajtj/c = z ∗ + ak1tk1/c + ak2tk2/c + ... (2)  \nj=1  \nwhere c ∈ N and 1 ≤ k1 \u003C k2 \u003C [k](k3... is)[3](k3... is)[...](k3... is)[ is](k3.","cbCaicpfjTssx9w3","https://ap.wps.com/l/cbCaicpfjTssx9w3","pdf",2279316,1,35,"English","en",105,"# Abstract\n# Introduction\n## Problem setup and homotopy tracking\n## The endgame regime near singular solutions\n## Corank-1 problems\n## Main theorem and algorithmic convergence","[{\"question\":\"Why does homotopy tracking become challenging for singular roots in Numerical Algebraic Geometry?\",\"answer\":\"Tracking the solution curve becomes especially difficult as the parameter approaches zero because the target solution is singular, meaning the Jacobian becomes singular. This creates the “endgame” regime where standard numerical behavior degrades.\"},{\"question\":\"What is the key idea of the proposed Arclength Endgame for corank-1 systems?\",\"answer\":\"The method combines classical pseudo-arclength continuation with estimation of the Puiseux series of the curve near the singular root. It achieves a proven superlinear convergence rate using only evaluations of the system and its Jacobian.\"},{\"question\":\"How does the work extend estimation of Puiseux series beyond the first exponent ratio?\",\"answer\":\"It introduces a new estimation method that is empirically more stable than prior approaches addressing only the first ratio k1/c. The paper also shows how to estimate ki/c for i ≥ 2.\"}]",1784192743,88,{"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},"computing-singular-solutions-of-polynomial-systems-towards-superlinear-convergence-without-deflation","",{"@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/computing-singular-solutions-of-polynomial-systems-towards-superlinear-convergence-without-deflation/84094/",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},"Why does homotopy tracking become challenging for singular roots in Numerical Algebraic Geometry?","Question",{"text":74,"@type":75},"Tracking the solution curve becomes especially difficult as the parameter approaches zero because the target solution is singular, meaning the Jacobian becomes singular. This creates the “endgame” regime where standard numerical behavior degrades.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"What is the key idea of the proposed Arclength Endgame for corank-1 systems?",{"text":79,"@type":75},"The method combines classical pseudo-arclength continuation with estimation of the Puiseux series of the curve near the singular root. It achieves a proven superlinear convergence rate using only evaluations of the system and its Jacobian.",{"name":81,"@type":72,"acceptedAnswer":82},"How does the work extend estimation of Puiseux series beyond the first exponent ratio?",{"text":83,"@type":75},"It introduces a new estimation method that is empirically more stable than prior approaches addressing only the first ratio k1/c. The paper also shows how to estimate ki/c for i ≥ 2.","https://schema.org",{"og:url":51,"og:type":86,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":88,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":91},[92,96,100,104,109,114,119,122,127,130,134],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":93,"show_sort_weight":94,"slug":95},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":97,"show_sort_weight":98,"slug":99},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":101,"show_sort_weight":102,"slug":103},"Exam",70,"exam",{"id":105,"doc_module":4,"doc_module_name":45,"category_name":106,"show_sort_weight":107,"slug":108},5,"Comic",60,"comic",{"id":110,"doc_module":4,"doc_module_name":45,"category_name":111,"show_sort_weight":112,"slug":113},6,"Technology",50,"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":125,"slug":126},9,"Religion & Spirituality",20,"religion-spirituality",{"id":125,"doc_module":4,"doc_module_name":45,"category_name":128,"show_sort_weight":125,"slug":129},"World Cup","world-cup",{"id":131,"doc_module":4,"doc_module_name":45,"category_name":132,"show_sort_weight":131,"slug":133},10,"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":45,"category_name":136,"show_sort_weight":105,"slug":137},19,"General","general"]