{"title":"Curves with few bad primes over cyclotomic ℤℓ-extensions","authors":"Samir Siksek, Robin Visser","doi":"10.2140/ant.2025.19.113","DOIUrl":"https://doi.org/10.2140/ant.2025.19.113","url":null,"abstract":"<p>Let <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>K</mi></math> be a number field, and <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>S</mi></math> a finite set of nonarchimedean places of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>K</mi></math>, and write <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi mathvariant=\"bold-script\">𝒪</mi></mrow><mrow><mo>×</mo></mrow></msup></math> for the group of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>S</mi></math>-units of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>K</mi></math>. A famous theorem of Siegel asserts that the <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>S</mi></math>-unit equation <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>𝜀</mi>\u0000<mo>+</mo>\u0000<mi>δ</mi>\u0000<mo>=</mo> <mn>1</mn></math>, with <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>𝜀</mi></math>, <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>δ</mi>\u0000<mo>∈</mo><msup><mrow><mi mathvariant=\"bold-script\">𝒪</mi></mrow><mrow><mo>×</mo></mrow></msup></math>, has only finitely many solutions. A famous theorem of Shafarevich asserts that there are only finitely many isomorphism classes of elliptic curves over <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>K</mi></math> with good reduction outside <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>S</mi></math>. Now instead of a number field, let <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>K</mi>\u0000<mo>=</mo> <msub><mrow><mi>ℚ</mi></mrow><mrow><mi>∞</mi><mo>,</mo><mi>ℓ</mi></mrow></msub></math> which denotes the <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>ℤ</mi></mrow><mrow><mi>ℓ</mi></mrow></msub></math>-cyclotomic extension of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>ℚ</mi></math>. We show that the <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>S</mi></math>-unit equation <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>𝜀</mi>\u0000<mo>+</mo>\u0000<mi>δ</mi>\u0000<mo>=</mo> <mn>1</mn></math>, with <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>𝜀</mi></math>, <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>δ</mi>\u0000<mo>∈</mo><msup><mrow><mi mathvariant=\"bold-script\">𝒪</mi></mrow><mrow><mo>×</mo></mrow></msup></math>, has infinitely many solutions for <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>ℓ</mi>\u0000<mo>∈</mo><mo stretchy=\"false\">{</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>5</mn><mo>,</mo><mn>7</mn><mo stretchy=\"false\">}</mo></math>, where <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>S</mi></math> consists only of the totally ramified prime above <math display=\"inline\" xmlns=\"http:","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"47 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142776370","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Super-Hölder vectors and the field of norms","authors":"Laurent Berger, Sandra Rozensztajn","doi":"10.2140/ant.2025.19.195","DOIUrl":"https://doi.org/10.2140/ant.2025.19.195","url":null,"abstract":"<p>Let <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>E</mi></math> be a field of characteristic <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math>. In a previous paper of ours, we defined and studied super-Hölder vectors in certain <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>E</mi></math>-linear representations of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>ℤ</mi></mrow><mrow><mi>p</mi></mrow></msub></math>. In the present paper, we define and study super-Hölder vectors in certain <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>E</mi></math>-linear representations of a general <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math>-adic Lie group. We then consider certain <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math>-adic Lie extensions <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>K</mi></mrow><mrow><mi>∞</mi></mrow></msub><mo>∕</mo><mi>K</mi></math> of a <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math>-adic field <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>K</mi></math>, and compute the super-Hölder vectors in the tilt of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>K</mi></mrow><mrow><mi>∞</mi></mrow></msub></math>. We show that these super-Hölder vectors are the perfection of the field of norms of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>K</mi></mrow><mrow><mi>∞</mi></mrow></msub><mo>∕</mo><mi>K</mi></math>. By specializing to the case of a Lubin–Tate extension, we are able to recover <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>E</mi><mo stretchy=\"false\">(</mo><mo stretchy=\"false\">(</mo><mi>Y</mi>\u0000<mo stretchy=\"false\">)</mo><mo stretchy=\"false\">)</mo></math> inside the <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>Y</mi> </math>-adic completion of its perfection, seen as a valued <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>E</mi></math>-vector space endowed with the action of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msubsup><mrow><mi mathvariant=\"bold-script\">𝒪</mi></mrow><mrow><mi>K</mi></mrow><mrow><mo>×</mo></mrow></msubsup></math> given by the endomorphisms of the corresponding Lubin–Tate group. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"20 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142776377","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Moduli of linear slices of high degree smooth hypersurfaces","authors":"Anand Patel, Eric Riedl, Dennis Tseng","doi":"10.2140/ant.2024.18.2133","DOIUrl":"https://doi.org/10.2140/ant.2024.18.2133","url":null,"abstract":"<p>We study the variation of linear sections of hypersurfaces in <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>ℙ</mi></mrow><mrow><mi>n</mi></mrow></msup></math>. We completely classify all plane curves, necessarily singular, whose line sections do not vary maximally in moduli. In higher dimensions, we prove that the family of hyperplane sections of any smooth degree <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>d</mi></math> hypersurface in <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>ℙ</mi></mrow><mrow><mi>n</mi></mrow></msup></math> varies maximally for <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>d</mi>\u0000<mo>≥</mo>\u0000<mi>n</mi>\u0000<mo>+</mo> <mn>3</mn></math>. In the process, we generalize the classical Grauert–Mülich theorem about lines in projective space, both to <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>k</mi></math>-planes in projective space and to free rational curves on arbitrary varieties. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"225 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142487002","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Separating G2-invariants of several octonions","authors":"Artem Lopatin, Alexandr N. Zubkov","doi":"10.2140/ant.2024.18.2157","DOIUrl":"https://doi.org/10.2140/ant.2024.18.2157","url":null,"abstract":"<p>We describe separating <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub></math>-invariants of several copies of the algebra of octonions over an algebraically closed field of characteristic two. We also obtain a minimal separating and a minimal generating set for <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub></math>-invariants of several copies of the algebra of octonions in case of a field of odd characteristic. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"25 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142486628","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Matrix Kloosterman sums","authors":"Márton Erdélyi, Árpád Tóth","doi":"10.2140/ant.2024.18.2247","DOIUrl":"https://doi.org/10.2140/ant.2024.18.2247","url":null,"abstract":"<p>We study a family of exponential sums that arises in the study of expanding horospheres on <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi> GL</mi><mo> <!--FUNCTION APPLICATION--> </mo><!--nolimits--></mrow><mrow><mi>n</mi></mrow></msub></math>. We prove an explicit version of general purity and find optimal bounds for these sums. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"210 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142487037","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Scattering diagrams for generalized cluster algebras","authors":"Lang Mou","doi":"10.2140/ant.2024.18.2179","DOIUrl":"https://doi.org/10.2140/ant.2024.18.2179","url":null,"abstract":"<p>We construct scattering diagrams for Chekhov–Shapiro generalized cluster algebras where exchange polynomials are factorized into binomials, generalizing the cluster scattering diagrams of Gross, Hacking, Keel and Kontsevich. They turn out to be natural objects arising in Fock and Goncharov’s cluster duality. Analogous features and structures (such as positivity and the cluster complex structure) in the ordinary case also appear in the generalized situation. With the help of these scattering diagrams, we show that generalized cluster variables are theta functions and hence have certain positivity property with respect to the coefficients in the binomial factors. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"66 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142486664","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Rooted tree maps for multiple L-values from a perspective of harmonic algebras","authors":"Hideki Murahara, Tatsushi Tanaka, Noriko Wakabayashi","doi":"10.2140/ant.2024.18.2003","DOIUrl":"https://doi.org/10.2140/ant.2024.18.2003","url":null,"abstract":"<p>We show the image of rooted tree maps forms a subspace of the kernel of the evaluation map of multiple <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>L</mi></math>-values. To prove this, we define the diamond product as a modified harmonic product and describe its properties. We also show that <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>τ</mi></math>-conjugate rooted tree maps are their antipodes. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"4 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142449537","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The distribution of large quadratic character sums and applications","authors":"Youness Lamzouri","doi":"10.2140/ant.2024.18.2091","DOIUrl":"https://doi.org/10.2140/ant.2024.18.2091","url":null,"abstract":"<p>We investigate the distribution of the maximum of character sums over the family of primitive quadratic characters attached to fundamental discriminants <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mo>|</mo><mi>d</mi><mo>|</mo><mo>≤</mo>\u0000<mi>x</mi></math>. In particular, our work improves results of Montgomery and Vaughan, and gives strong evidence that the Omega result of Bateman and Chowla for quadratic character sums is optimal. We also obtain similar results for real characters with prime discriminants up to <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>x</mi></math>, and deduce the interesting consequence that almost all primes with large Legendre symbol sums are congruent to <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>3</mn></math> modulo <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>4</mn></math>. Our results are motivated by a recent work of Bober, Goldmakher, Granville and Koukoulopoulos, who proved similar results for the family of nonprincipal characters modulo a large prime. However, their method does not seem to generalize to other families of Dirichlet characters. Instead, we use a different and more streamlined approach, which relies mainly on the quadratic large sieve. As an application, we consider a question of Montgomery concerning the positivity of sums of Legendre symbols. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"17 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142449539","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Terminal orders on arithmetic surfaces","authors":"Daniel Chan, Colin Ingalls","doi":"10.2140/ant.2024.18.2027","DOIUrl":"https://doi.org/10.2140/ant.2024.18.2027","url":null,"abstract":"<p>The local structure of terminal Brauer classes on arithmetic surfaces was classified (2021), generalising the classification on geometric surfaces (2005). Part of the interest in these classifications is that it enables the minimal model program to be applied to the noncommutative setting of orders on surfaces. We give étale local structure theorems for terminal orders on arithmetic surfaces, at least when the degree is a prime <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi>\u0000<mo>></mo> <mn>5</mn></math>. This generalises the structure theorem given in the geometric case. They can all be explicitly constructed as algebras of matrices over symbols. From this description one sees that such terminal orders all have global dimension two, thus generalising the fact that terminal (commutative) surfaces are smooth and hence homologically regular. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"109 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142449538","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Word measures on GLn(q) and free group algebras","authors":"Danielle Ernst-West, Doron Puder, Matan Seidel","doi":"10.2140/ant.2024.18.2047","DOIUrl":"https://doi.org/10.2140/ant.2024.18.2047","url":null,"abstract":"<p>Fix a finite field <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>K</mi></math> of order <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>q</mi></math> and a word <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>w</mi></math> in a free group <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mstyle mathvariant=\"bold-italic\"><mi>F</mi></mstyle></math> on <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>r</mi></math> generators. A <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>w</mi></math>-random element in <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi> GL</mi><mo> <!--FUNCTION APPLICATION--> </mo><!--nolimits--></mrow><mrow><mi>N</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>K</mi><mo stretchy=\"false\">)</mo></math> is obtained by sampling <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>r</mi></math> independent uniformly random elements <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>g</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mi>…</mi><mo> <!--FUNCTION APPLICATION--></mo><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>r</mi></mrow></msub>\u0000<mo>∈</mo><msub><mrow><mi> GL</mi><mo> <!--FUNCTION APPLICATION--> </mo><!--nolimits--></mrow><mrow><mi>N</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>K</mi><mo stretchy=\"false\">)</mo></math> and evaluating <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>w</mi><mo stretchy=\"false\">(</mo><msub><mrow><mi>g</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mi>…</mi><mo> <!--FUNCTION APPLICATION--></mo><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>r</mi></mrow></msub><mo stretchy=\"false\">)</mo></math>. Consider <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi mathvariant=\"double-struck\">𝔼</mi></mrow><mrow><mi>w</mi></mrow></msub><mo stretchy=\"false\">[</mo><mi>fix</mi><mo> <!--FUNCTION APPLICATION--> </mo><!--nolimits--><mo stretchy=\"false\">]</mo></math>, the average number of vectors in <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>K</mi></mrow><mrow><mi>N</mi></mrow></msup></math> fixed by a <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>w</mi></math>-random element. We show that <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi mathvariant=\"double-struck\">𝔼</mi></mrow><mrow><mi>w</mi></mrow></msub><mo stretchy=\"false\">[</mo><mi>fix</mi><mo> <!--FUNCTION APPLICATION--> </mo><!--nolimits--><mo stretchy=\"false\">]</mo></math> is a rational function in <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>q</mi></mrow><mrow><mi>N</mi></mrow></msup></math>. If <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>w</mi>\u0000<mo>=</mo> <msup><mrow><mi>u</mi></mrow><mrow><mi>d</mi></mrow></ms","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"46 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142449570","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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