Algebra & Number Theory最新文献

{"title":"A case study of intersections on blowups of the moduli of curves","authors":"Sam Molcho, Dhruv Ranganathan","doi":"10.2140/ant.2024.18.1767","DOIUrl":"https://doi.org/10.2140/ant.2024.18.1767","url":null,"abstract":"<p>We explain how logarithmic structures select principal components in an intersection of schemes. These manifest in Chow homology and can be understood using strict transforms under logarithmic blowups. Our motivation comes from Gromov–Witten theory. The <span>toric contact cycles </span>in the moduli space of curves parameterize curves that admit a map to a fixed toric variety with prescribed contact orders. We show that they are intersections of virtual strict transforms of double ramification cycles in blowups of the moduli space of curves. We supply a calculation scheme for the virtual strict transforms, and deduce that toric contact cycles lie in the tautological ring of the moduli space of curves. This is a higher-dimensional analogue of a result of Faber and Pandharipande. The operational Chow rings of Artin fans play a basic role, and are shown to be isomorphic to rings of piecewise polynomials on associated cone complexes. The ingredients in our analysis are Fulton’s blowup formula, Aluffi’s formulas for Segre classes of monomial schemes, piecewise polynomials, and degeneration methods. A model calculation in toric intersection theory is treated without logarithmic methods and may be read independently. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"31 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142384206","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":"Spectral moment formulae for GL(3) × GL(2) L-functions I : The cuspidal case","authors":"Chung-Hang Kwan","doi":"10.2140/ant.2024.18.1817","DOIUrl":"https://doi.org/10.2140/ant.2024.18.1817","url":null,"abstract":"<p>Spectral moment formulae of various shapes have proven very successful in studying the statistics of central <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>L</mi></math>-values. We establish, in a completely explicit fashion, such formulae for the family of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi> GL</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--><mo stretchy=\"false\">(</mo><mn>3</mn><mo stretchy=\"false\">)</mo>\u0000<mo>×</mo><mi> GL</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--><mo stretchy=\"false\">(</mo><mn>2</mn><mo stretchy=\"false\">)</mo></math> Rankin–Selberg <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>L</mi></math>-functions using the period integral method. Our argument does not rely on either the Kuznetsov or Voronoi formulae. We also prove the essential analytic properties and derive explicit formulae for the integral transform of our moment formulae. We hope that our method will provide deeper insights into moments of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>L</mi></math>-functions for higher-rank groups. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"12 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142384207","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":"A geometric classification of the holomorphic vertex operator algebras of central charge 24","authors":"Sven Möller, Nils R. Scheithauer","doi":"10.2140/ant.2024.18.1891","DOIUrl":"https://doi.org/10.2140/ant.2024.18.1891","url":null,"abstract":"<p>We associate with a generalised deep hole of the Leech lattice vertex operator algebra a generalised hole diagram. We show that this Dynkin diagram determines the generalised deep hole up to conjugacy and that there are exactly <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>7</mn><mn>0</mn></math> such diagrams. In an earlier work we proved a bijection between the generalised deep holes and the strongly rational, holomorphic vertex operator algebras of central charge <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>2</mn><mn>4</mn></math> with nontrivial weight-<math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>1</mn></math> space. Hence, we obtain a new, geometric classification of these vertex operator algebras, generalising the classification of the Niemeier lattices by their hole diagrams. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"10 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142384211","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 wavefront sets of unipotent supercuspidal representations","authors":"Dan Ciubotaru, Lucas Mason-Brown, Emile Okada","doi":"10.2140/ant.2024.18.1863","DOIUrl":"https://doi.org/10.2140/ant.2024.18.1863","url":null,"abstract":"<p>We prove that the double (or canonical unramified) wavefront set of an irreducible depth-0 supercuspidal representation of a reductive <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math>-adic group is a singleton provided <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi>\u0000<mo>&gt;</mo> <mn>3</mn><mo stretchy=\"false\">(</mo><mi>h</mi>\u0000<mo>−</mo> <mn>1</mn><mo stretchy=\"false\">)</mo></math>, where <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>h</mi></math> is the Coxeter number. We deduce that the geometric wavefront set is also a singleton in this case, proving a conjecture of Mœglin and Waldspurger. When the group is inner to split and the representation belongs to Lusztig’s category of unipotent representations, we give an explicit formula for the double and geometric wavefront sets. As a consequence, we show that the nilpotent part of the Deligne–Langlands–Lusztig parameter of a unipotent supercuspidal representation is precisely the image of its geometric wavefront set under Spaltenstein’s duality map. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"29 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142384208","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":"Affine Deligne–Lusztig varieties with finite Coxeter parts","authors":"Xuhua He, Sian Nie, Qingchao Yu","doi":"10.2140/ant.2024.18.1681","DOIUrl":"https://doi.org/10.2140/ant.2024.18.1681","url":null,"abstract":"<p>We study affine Deligne–Lusztig varieties <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>X</mi></mrow><mrow><mi>w</mi><mo stretchy=\"false\">(</mo><mi>b</mi><mo stretchy=\"false\">)</mo></mrow></msub></math> when the finite part of the element <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>w</mi></math> in the Iwahori–Weyl group is a partial <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>σ</mi></math>-Coxeter element. We show that such <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>w</mi></math> is a cordial element and <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>X</mi></mrow><mrow><mi>w</mi><mo stretchy=\"false\">(</mo><mi>b</mi><mo stretchy=\"false\">)</mo></mrow></msub><mo>≠</mo><mi>∅</mi></math> if and only if <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>b</mi></math> satisfies a certain Hodge–Newton indecomposability condition. Our main result is that for such <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>w</mi></math> and <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>b</mi></math>, <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>X</mi></mrow><mrow><mi>w</mi><mo stretchy=\"false\">(</mo><mi>b</mi><mo stretchy=\"false\">)</mo></mrow></msub></math> has a simple geometric structure: the <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>σ</mi></math>-centralizer of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>b</mi></math> acts transitively on the set of irreducible components of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>X</mi></mrow><mrow><mi>w</mi><mo stretchy=\"false\">(</mo><mi>b</mi><mo stretchy=\"false\">)</mo></mrow></msub></math>; and each irreducible component is an iterated fibration over a classical Deligne–Lusztig variety of Coxeter type, and the iterated fibers are either <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi mathvariant=\"double-struck\">𝔸</mi></mrow><mrow><mn>1</mn></mrow></msup></math> or <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi mathvariant=\"double-struck\">𝔾</mi></mrow><mrow><mi>m</mi></mrow></msub></math>. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"9 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142245248","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":"A unipotent realization of the chromatic quasisymmetric function","authors":"Lucas Gagnon","doi":"10.2140/ant.2024.18.1737","DOIUrl":"https://doi.org/10.2140/ant.2024.18.1737","url":null,"abstract":"<p>We realize two families of combinatorial symmetric functions via the complex character theory of the finite general linear group <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><msub><mrow><mi mathvariant=\"double-struck\">𝔽</mi></mrow><mrow><mi>q</mi></mrow></msub><mo stretchy=\"false\">)</mo></math>: chromatic quasisymmetric functions and vertical strip LLT polynomials. The associated <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><msub><mrow><mi mathvariant=\"double-struck\">𝔽</mi></mrow><mrow><mi>q</mi></mrow></msub><mo stretchy=\"false\">)</mo></math> characters are elementary in nature and can be obtained by induction from certain well-behaved characters of the unipotent upper triangular groups <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi> UT</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy=\"false\">(</mo><msub><mrow><mi mathvariant=\"double-struck\">𝔽</mi></mrow><mrow><mi>q</mi></mrow></msub><mo stretchy=\"false\">)</mo></math>. The proof of these results also gives a general Hopf algebraic approach to computing the induction map. Additional results include a connection between the relevant <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><msub><mrow><mi mathvariant=\"double-struck\">𝔽</mi></mrow><mrow><mi>q</mi></mrow></msub><mo stretchy=\"false\">)</mo></math> characters and Hessenberg varieties and a reinterpretation of known theorems and conjectures about the relevant symmetric functions in terms of <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><msub><mrow><mi mathvariant=\"double-struck\">𝔽</mi></mrow><mrow><mi>q</mi></mrow></msub><mo stretchy=\"false\">)</mo></math>. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"112 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142245253","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":"A bound for the exterior product of S-units","authors":"Shabnam Akhtari, Jeffrey D. Vaaler","doi":"10.2140/ant.2024.18.1589","DOIUrl":"https://doi.org/10.2140/ant.2024.18.1589","url":null,"abstract":"<p>We generalize an inequality for the determinant of a real matrix proved by A. Schinzel, to more general exterior products of vectors in Euclidean space. We apply this inequality to the logarithmic embedding of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>S</mi></math>-units contained in a number field <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>k</mi></math>. This leads to a bound for the exterior product of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>S</mi></math>-units expressed as a product of heights. Using a volume formula of P. McMullen we show that our inequality is sharp up to a constant that depends only on the rank of the <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>S</mi></math>-unit group but not on the field <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>k</mi></math>. Our inequality is related to a conjecture of F. Rodriguez Villegas. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"64 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142245251","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":"Prime values of f(a,b2) and f(a,p2), f quadratic","authors":"Stanley Yao Xiao","doi":"10.2140/ant.2024.18.1619","DOIUrl":"https://doi.org/10.2140/ant.2024.18.1619","url":null,"abstract":"<p>We prove an asymptotic formula for primes of the shape <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>f</mi><mo stretchy=\"false\">(</mo><mi>a</mi><mo>,</mo><msup><mrow><mi>b</mi></mrow><mrow><mn>2</mn></mrow></msup><mo stretchy=\"false\">)</mo></math> with <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>a</mi></math>, <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>b</mi></math> integers and of the shape <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>f</mi><mo stretchy=\"false\">(</mo><mi>a</mi><mo>,</mo><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msup><mo stretchy=\"false\">)</mo></math> with <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math> prime. Here <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>f</mi></math> is a binary quadratic form with integer coefficients, irreducible over <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>ℚ</mi></math> and has no local obstructions. This refines the seminal work of Friedlander and Iwaniec on primes of the form <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup>\u0000<mo>+</mo> <msup><mrow><mi>y</mi></mrow><mrow><mn>4</mn></mrow></msup></math> and of Heath-Brown and Li on primes of the form <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msup>\u0000<mo>+</mo> <msup><mrow><mi>p</mi></mrow><mrow><mn>4</mn></mrow></msup></math>, as well as earlier work of the author with Lam and Schindler on primes of the form <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>f</mi><mo stretchy=\"false\">(</mo><mi>a</mi><mo>,</mo><mi>p</mi><mo stretchy=\"false\">)</mo></math> with <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>f</mi></math> a positive definite form. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"12 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142245246","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":"Semistable models for some unitary Shimura varieties over ramified primes","authors":"Ioannis Zachos","doi":"10.2140/ant.2024.18.1715","DOIUrl":"https://doi.org/10.2140/ant.2024.18.1715","url":null,"abstract":"<p>We consider Shimura varieties associated to a unitary group of signature <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mo stretchy=\"false\">(</mo><mi>n</mi>\u0000<mo>−</mo> <mn>2</mn><mo>,</mo><mn>2</mn><mo stretchy=\"false\">)</mo></math>. We give regular <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math>-adic integral models for these varieties over odd primes <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math> which ramify in the imaginary quadratic field with level subgroup at <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math> given by the stabilizer of a selfdual lattice in the hermitian space. Our construction is given by an explicit resolution of a corresponding local model. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"197 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142245250","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":"Application of a polynomial sieve: beyond separation of variables","authors":"Dante Bonolis, Lillian B. Pierce","doi":"10.2140/ant.2024.18.1515","DOIUrl":"https://doi.org/10.2140/ant.2024.18.1515","url":null,"abstract":"&lt;p&gt;Let a polynomial &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt;f&lt;/mi&gt;\u0000&lt;mo&gt;∈&lt;/mo&gt;\u0000&lt;mi&gt;ℤ&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;[&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;…&lt;/mi&gt;&lt;mo&gt; ⁡&lt;!--FUNCTION APPLICATION--&gt;&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=\"false\"&gt;]&lt;/mo&gt;&lt;/math&gt; be given. The square sieve can provide an upper bound for the number of integral &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mstyle mathvariant=\"bold-italic\"&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mstyle&gt;\u0000&lt;mo&gt;∈&lt;/mo&gt; &lt;msup&gt;&lt;mrow&gt;&lt;mo stretchy=\"false\"&gt;[&lt;/mo&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt; such that &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;mstyle mathvariant=\"bold-italic\"&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mstyle&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;/math&gt; is a perfect square. Recently this has been generalized substantially: first to a power sieve, counting &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mstyle mathvariant=\"bold-italic\"&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mstyle&gt;\u0000&lt;mo&gt;∈&lt;/mo&gt; &lt;msup&gt;&lt;mrow&gt;&lt;mo stretchy=\"false\"&gt;[&lt;/mo&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt; for which &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;mstyle mathvariant=\"bold-italic\"&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mstyle&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;\u0000&lt;mo&gt;=&lt;/mo&gt; &lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt; is solvable for &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt;y&lt;/mi&gt;\u0000&lt;mo&gt;∈&lt;/mo&gt;\u0000&lt;mi&gt;ℤ&lt;/mi&gt;&lt;/math&gt;; then to a polynomial sieve, counting &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mstyle mathvariant=\"bold-italic\"&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mstyle&gt;\u0000&lt;mo&gt;∈&lt;/mo&gt; &lt;msup&gt;&lt;mrow&gt;&lt;mo stretchy=\"false\"&gt;[&lt;/mo&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt; for which &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;mstyle mathvariant=\"bold-italic\"&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mstyle&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;\u0000&lt;mo&gt;=&lt;/mo&gt;\u0000&lt;mi&gt;g&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;/math&gt; is solvable, for a given polynomial &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;/math&gt;. Formally, a polynomial sieve lemma can encompass the more general problem of counting &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mstyle mathvariant=\"bold-italic\"&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mstyle&gt;\u0000&lt;mo&gt;∈&lt;/mo&gt; &lt;msup&gt;&lt;mrow&gt;&lt;mo stretchy=\"false\"&gt;[&lt;/mo&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt; for which &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mstyle mathvariant=\"bold-italic\"&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mstyle&gt;&lt;mo stre","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"12 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142236172","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}