[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83680-en":3,"doc-seo-83680-105":30,"detail-sidebar-cat-0-en-105":95},{"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},83680,4810365810221,"Aurora","https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d",8,"Research & Report","On Factoring Quantum-Plane Skew Polynomials over Q(ω)(t)","Algorithms for factoring dilation skew polynomials in the quantum-plane (dilation) setting are developed over a function field tied to a cyclotomic field. The algebra R=K(σ)[δ;σ] with K=Q(ω) is analyzed using central elements to factor central left multiples and obtain a partial decomposition in characteristic zero. A two-level modular strategy specializes the central parameter to good values, studies cyclic algebras over number fields, then reduces at inert primes to apply fast finite-field skew-factorization and lift factors back. SageMath prototypes are provided, and extending constants to Q enables decidable factorization in an exact algebraic model.","arXiv :2607 .0275 1v 1 [ cs . SC] 2 Jul 2026  \nOn Factoring Quantum-Plane Skew Polynomials  \nover Q (􀀸)(􀁴)  \nMark Giesbrecht  \nCheriton School of Computer Science, Faculty of Mathematics, University of Waterloo, Canada  \n[mwg@uwaterloo.ca](mwg@uwaterloo.ca)  \nAbstract. We study algorithms for factorization in the quantum plane of (dilation) skew polynomials over a function field of a cyclotomic field:  \nR = K (􀁃)[􀁇; 􀁦], K = Q (􀁬), 􀁦 (􀁃) = 􀁬􀁃,  \nwhere 􀁬 ∈ C is a primitive 􀀼-th root of unity. We start with the established approach through central elements and factor the central left multiples, staying in characteristic zero, to obtain a partial decomposition. A two-level modular approach is proposed: specialize a central parameter to good algebraic values, study the resulting cyclic algebras over number fields, and then reduce further at good inert primes so that fast finite-field skew-factorization algorithms apply. A prototype SageMath implementation is provided to experiment with the algorithms. We then look at the effect of extending the field of constants from Q (􀁬) to Q, an algebraic closure of Q, and factoring over Q (􀁃)[􀁇; 􀁦] . In this case we show factorization is decidable in the exact algebraic model based on finite extensions.  \n1 Introduction  \nThis paper studies factorization of skew polynomials over a function field over a cyclotomic field with a dilation automorphism. This is a characteristic-zero analogue of the Ore polynomial setting over F􀁀 (􀁃) that underlies earlier skew factorization algorithms. In the root-of-unity quantum-plane case, the large univariate centre makes a “bound first” approach effective.  \nLet K = Q (􀁬), where 􀁬 ∈ C is a primitive 􀀼-th root of unity, and let R = K (􀁃)[􀁇; 􀁦], 􀁇􀀰 = 􀁦 (􀀰)􀁇 (for any 􀀰 ∈ K (􀁃)), 􀁦(􀁃) = 􀁬􀁃 .  \nBecause 􀁦 has finite order 􀀼, its fixed field is K (􀁃􀀼 ) and the centre of R is the commutative principal ideal domain  \nC = K (􀁃􀀼 )[􀁇􀀼 ] = K(T )[X], T = 􀁃􀀼 , X = 􀁇􀀼 .  \nThe centre is a polynomial ring in the single central variable X over the rational function field K(T ) .  \nTo appear, Computer Algebra in Scientific Computation (CASC) conference, August 31– September 4, 2026, Bath, UK  \n2 Mark Giesbrecht  \nFor our algorithms the initial input should first be cleared of denominators, but the factorization stages are carried out for a monic associate. Let 􀀛in ∈ R \\ {0} be the given input. Clearing denominators and removing scalar content chooses an integral representative 􀀛 ∈ K [􀁃][􀁇; 􀁦] and a unit 􀁗 ∈ K (􀁃)× with 􀀛 = 􀁗􀀛in . Put ℓ = lc 􀁇 (􀀛) and 􀀵 = 􀀛♯ := ℓ −1􀀛 . Then 􀀵 is monic and 􀀛in = 􀁗 −1ℓ 􀀵 . The scalar 􀁗−1ℓ is a unit of R, so 􀀛in and 􀀵 have the same right divisors and the same irreducibility status, up to multiplication by units. Below, unless explicitly stated otherwise, 􀀵 denotes this working monic associate. A factorization  \n􀀵 = 􀀵 1 􀀵 2 · · · 􀀵 􀀺  \ninto irreducibles in R gives the corresponding factorization  \n􀀛in = 􀁗 −1ℓ 􀀵 1 􀀵 2 · · · 􀀵 􀀺 .  \nDenominators in 􀁃 may necessarily occur in the factors even when the chosen integral representative 􀀛 has coefficients in K [􀁃] .  \nWe use right-divisibility conventions throughout: ℎ is a right divisor of 􀀶 if 􀀶 = 􀁀ℎ . We write gcrd (􀀰, 􀀱) for the monic greatest common right divisor, equivalently the monic generator of R􀀰 + R􀀱 . For ℎ ≠ 0, rrem (􀀶, ℎ) denotes the right remainder in the division 􀀶 = 􀁀ℎ + 􀁁, where 􀁁 = 0 or deg 􀁇􀁁 \u003C deg 􀁇 ℎ .  \nA nonzero 􀀵 ∈ R is bounded if R 􀀵 ∩ C ≠ 0, i.e., if 􀀵 has a nonzero central left multiple. This is automatic in R, since it is free over C and is a domain. Following Jacobson (1943), the bound of 􀀵 is the unique monic 􀁩 ∈ C ∩ R 􀀵 of minimal degree in X, say 􀁩 = 􀁄 􀀵 with 􀁄 ∈ R. Such a bound is unique and computable by linear algebra.  \nThe overall algorithmic approach initially follows that of Giesbrecht (1998) and subsequent algorithms. First we compute a monic bound 􀁩 ∈ C. If 􀁩 factors in C, then right gcd computations split 􀀵 into a rough factorization whose pieces have irreducible ce","cbCaic19SPjFDrCd","https://ap.wps.com/l/cbCaic19SPjFDrCd","pdf",428465,4,1,20,"English","en",105,"# Introduction\n## Algebraic setup and central structure\n## Algorithmic approach and modular strategy\n## Constant-field extensions and decidability\n# Related work and further sections","[{\"question\":\"What mathematical object is targeted for factorization in the document?\",\"answer\":\"Factorization is studied for dilation skew polynomials in a quantum-plane algebra of the form R over a function field associated with a cyclotomic field, with K=Q(ω) and a finite-order automorphism σ.\"},{\"question\":\"How does the method use the centre to simplify the factorization task?\",\"answer\":\"The algorithm first computes a monic bound in the commutative centre, then factors central left multiples and recursively handles multiplicities of central factors, reducing the hard remaining case to an irreducible central-bound situation.\"},{\"question\":\"What is the two-level modular approach and why is it used?\",\"answer\":\"The approach performs characteristic-zero computations long enough to exploit centre and cyclic-algebra structure, then specializes the central parameter to good algebraic values to descend to cyclic algebras over number fields. It further reduces at good inert primes so finite-field skew-factorization algorithms apply, after which factors are lifted back.\"},{\"question\":\"What changes when enlarging constants from K to Q?\",\"answer\":\"Stronger decidability results arise for a different factorization problem: in an algebraically closed constant-field setting, the irreducible-bound algebra splits by Tsen’s theorem, leading to decidable full factorization in an exact algebraic model based on finite extensions and root adjunction.\"}]",1784189704,50,{"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":90,"head_meta":92,"extra_data":94,"updated_unix":28},"on-factoring-quantum-plane-skew-polynomials-over-qt","",{"@graph":36,"@context":89},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"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":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":20},"https://docshare.wps.com/document/on-factoring-quantum-plane-skew-polynomials-over-qt/83680/",{"url":52,"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","2026-07-26","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81,85],{"name":72,"@type":73,"acceptedAnswer":74},"What mathematical object is targeted for factorization in the document?","Question",{"text":75,"@type":76},"Factorization is studied for dilation skew polynomials in a quantum-plane algebra of the form R over a function field associated with a cyclotomic field, with K=Q(ω) and a finite-order automorphism σ.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the method use the centre to simplify the factorization task?",{"text":80,"@type":76},"The algorithm first computes a monic bound in the commutative centre, then factors central left multiples and recursively handles multiplicities of central factors, reducing the hard remaining case to an irreducible central-bound situation.",{"name":82,"@type":73,"acceptedAnswer":83},"What is the two-level modular approach and why is it used?",{"text":84,"@type":76},"The approach performs characteristic-zero computations long enough to exploit centre and cyclic-algebra structure, then specializes the central parameter to good algebraic values to descend to cyclic algebras over number fields. It further reduces at good inert primes so finite-field skew-factorization algorithms apply, after which factors are lifted back.",{"name":86,"@type":73,"acceptedAnswer":87},"What changes when enlarging constants from K to Q?",{"text":88,"@type":76},"Stronger decidability results arise for a different factorization problem: in an algebraically closed constant-field setting, the irreducible-bound algebra splits by Tsen’s theorem, leading to decidable full factorization in an exact algebraic model based on finite extensions and root adjunction.","https://schema.org",{"og:url":52,"og:type":91,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":93,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":96},[97,101,105,109,114,118,123,126,130,133,137],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Literature",80,"literature",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":106,"show_sort_weight":107,"slug":108},"Exam",70,"exam",{"id":110,"doc_module":4,"doc_module_name":46,"category_name":111,"show_sort_weight":112,"slug":113},5,"Comic",60,"comic",{"id":115,"doc_module":4,"doc_module_name":46,"category_name":116,"show_sort_weight":29,"slug":117},6,"Technology","technology",{"id":119,"doc_module":4,"doc_module_name":46,"category_name":120,"show_sort_weight":121,"slug":122},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":124,"slug":125},30,"research-report",{"id":127,"doc_module":4,"doc_module_name":46,"category_name":128,"show_sort_weight":22,"slug":129},9,"Religion & Spirituality","religion-spirituality",{"id":22,"doc_module":4,"doc_module_name":46,"category_name":131,"show_sort_weight":22,"slug":132},"World Cup","world-cup",{"id":134,"doc_module":4,"doc_module_name":46,"category_name":135,"show_sort_weight":134,"slug":136},10,"Lifestyle","lifestyle",{"id":138,"doc_module":4,"doc_module_name":46,"category_name":139,"show_sort_weight":110,"slug":140},19,"General","general"]