Prime Number Theorems for Polynomials from Homogeneous Dynamics

IF 2.5 1区 数学 Q1 MATHEMATICS
Giorgos Kotsovolis, Katharine Woo
{"title":"Prime Number Theorems for Polynomials from Homogeneous Dynamics","authors":"Giorgos Kotsovolis, Katharine Woo","doi":"10.1007/s00039-025-00716-y","DOIUrl":null,"url":null,"abstract":"<p>We establish a new class of examples of the multivariate Bateman-Horn conjecture by using tools from dynamics. These cases include the determinant polynomial on the space of <span><span style=\"\"></span><span data-mathml='&lt;math xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;' role=\"presentation\" style=\"font-size: 100%; display: inline-block; position: relative;\" tabindex=\"0\"><svg aria-hidden=\"true\" focusable=\"false\" height=\"1.509ex\" role=\"img\" style=\"vertical-align: -0.205ex;\" viewbox=\"0 -561.7 2423.9 649.8\" width=\"5.63ex\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g fill=\"currentColor\" stroke=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use x=\"0\" xlink:href=\"#MJMATHI-6E\" y=\"0\"></use><use x=\"822\" xlink:href=\"#MJMAIN-D7\" y=\"0\"></use><use x=\"1823\" xlink:href=\"#MJMATHI-6E\" y=\"0\"></use></g></svg><span role=\"presentation\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>n</mi><mo>×</mo><mi>n</mi></math></span></span><script type=\"math/tex\">n\\times n</script></span> matrices, the Pfaffian on the space of skew-symmetric <span><span style=\"\"></span><span data-mathml='&lt;math xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;' role=\"presentation\" style=\"font-size: 100%; display: inline-block; position: relative;\" tabindex=\"0\"><svg aria-hidden=\"true\" focusable=\"false\" height=\"1.909ex\" role=\"img\" style=\"vertical-align: -0.205ex;\" viewbox=\"0 -733.9 3424.9 822.1\" width=\"7.955ex\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g fill=\"currentColor\" stroke=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use x=\"0\" xlink:href=\"#MJMAIN-32\" y=\"0\"></use><use x=\"500\" xlink:href=\"#MJMATHI-6E\" y=\"0\"></use><use x=\"1323\" xlink:href=\"#MJMAIN-D7\" y=\"0\"></use><use x=\"2323\" xlink:href=\"#MJMAIN-32\" y=\"0\"></use><use x=\"2824\" xlink:href=\"#MJMATHI-6E\" y=\"0\"></use></g></svg><span role=\"presentation\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>2</mn><mi>n</mi><mo>×</mo><mn>2</mn><mi>n</mi></math></span></span><script type=\"math/tex\">2n\\times 2n</script></span> matrices, and the determinant polynomial on the space of symmetric <span><span style=\"\"></span><span data-mathml='&lt;math xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;' role=\"presentation\" style=\"font-size: 100%; display: inline-block; position: relative;\" tabindex=\"0\"><svg aria-hidden=\"true\" focusable=\"false\" height=\"1.512ex\" role=\"img\" style=\"vertical-align: -0.205ex;\" viewbox=\"0 -562.7 2423.9 650.9\" width=\"5.63ex\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g fill=\"currentColor\" stroke=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use x=\"0\" xlink:href=\"#MJMATHI-6E\" y=\"0\"></use><use x=\"822\" xlink:href=\"#MJMAIN-D7\" y=\"0\"></use><use x=\"1823\" xlink:href=\"#MJMATHI-6E\" y=\"0\"></use></g></svg><span role=\"presentation\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>n</mi><mo>×</mo><mi>n</mi></math></span></span><script type=\"math/tex\">n\\times n</script></span> matrices. In particular, let <span><span style=\"\"></span><span data-mathml='&lt;math xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;/math&gt;' role=\"presentation\" style=\"font-size: 100%; display: inline-block; position: relative;\" tabindex=\"0\"><svg aria-hidden=\"true\" focusable=\"false\" height=\"2.614ex\" role=\"img\" style=\"vertical-align: -0.706ex;\" viewbox=\"0 -821.4 2743.2 1125.3\" width=\"6.371ex\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g fill=\"currentColor\" stroke=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use x=\"0\" xlink:href=\"#MJMAIN-28\" y=\"0\"></use><use x=\"389\" xlink:href=\"#MJMATHI-56\" y=\"0\"></use><use x=\"1159\" xlink:href=\"#MJMAIN-2C\" y=\"0\"></use><use x=\"1604\" xlink:href=\"#MJMATHI-46\" y=\"0\"></use><use x=\"2353\" xlink:href=\"#MJMAIN-29\" y=\"0\"></use></g></svg><span role=\"presentation\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mo stretchy=\"false\">(</mo><mi>V</mi><mo>,</mo><mi>F</mi><mo stretchy=\"false\">)</mo></math></span></span><script type=\"math/tex\">(V,F)</script></span> be any pair among the following: <span><span style=\"\"></span><span data-mathml='&lt;math xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi mathvariant=\"normal\"&gt;M&lt;/mi&gt;&lt;mi mathvariant=\"normal\"&gt;a&lt;/mi&gt;&lt;mi mathvariant=\"normal\"&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo movablelimits=\"true\" form=\"prefix\"&gt;det&lt;/mo&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;/math&gt;' role=\"presentation\" style=\"font-size: 100%; display: inline-block; position: relative;\" tabindex=\"0\"><svg aria-hidden=\"true\" focusable=\"false\" height=\"2.614ex\" role=\"img\" style=\"vertical-align: -0.706ex;\" viewbox=\"0 -821.4 4946.8 1125.3\" width=\"11.489ex\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g fill=\"currentColor\" stroke=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use x=\"0\" xlink:href=\"#MJMAIN-28\" y=\"0\"></use><g transform=\"translate(389,0)\"><use x=\"0\" xlink:href=\"#MJMAIN-4D\" y=\"0\"></use><use x=\"917\" xlink:href=\"#MJMAIN-61\" y=\"0\"></use><use x=\"1418\" xlink:href=\"#MJMAIN-74\" y=\"0\"></use><use transform=\"scale(0.707)\" x=\"2556\" xlink:href=\"#MJMATHI-6E\" y=\"-213\"></use></g><use x=\"2721\" xlink:href=\"#MJMAIN-2C\" y=\"0\"></use><g transform=\"translate(3166,0)\"><use xlink:href=\"#MJMAIN-64\"></use><use x=\"556\" xlink:href=\"#MJMAIN-65\" y=\"0\"></use><use x=\"1001\" xlink:href=\"#MJMAIN-74\" y=\"0\"></use></g><use x=\"4557\" xlink:href=\"#MJMAIN-29\" y=\"0\"></use></g></svg><span role=\"presentation\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mo stretchy=\"false\">(</mo><msub><mrow><mi mathvariant=\"normal\">M</mi><mi mathvariant=\"normal\">a</mi><mi mathvariant=\"normal\">t</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mo form=\"prefix\" movablelimits=\"true\">det</mo><mo stretchy=\"false\">)</mo></math></span></span><script type=\"math/tex\">(\\textrm{Mat}_{n}, \\det )</script></span>, <span><span style=\"\"></span><span data-mathml='&lt;math xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi mathvariant=\"normal\"&gt;S&lt;/mi&gt;&lt;mi mathvariant=\"normal\"&gt;k&lt;/mi&gt;&lt;mi mathvariant=\"normal\"&gt;e&lt;/mi&gt;&lt;mi mathvariant=\"normal\"&gt;w&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mrow&gt;&lt;mi mathvariant=\"normal\"&gt;P&lt;/mi&gt;&lt;mi mathvariant=\"normal\"&gt;f&lt;/mi&gt;&lt;mi mathvariant=\"normal\"&gt;f&lt;/mi&gt;&lt;/mrow&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;/math&gt;' role=\"presentation\" style=\"font-size: 100%; display: inline-block; position: relative;\" tabindex=\"0\"><svg aria-hidden=\"true\" focusable=\"false\" height=\"2.614ex\" role=\"img\" style=\"vertical-align: -0.706ex;\" viewbox=\"0 -821.4 5781.2 1125.3\" width=\"13.427ex\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g fill=\"currentColor\" stroke=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use x=\"0\" xlink:href=\"#MJMAIN-28\" y=\"0\"></use><g transform=\"translate(389,0)\"><use x=\"0\" xlink:href=\"#MJMAIN-53\" y=\"0\"></use><use x=\"556\" xlink:href=\"#MJMAIN-6B\" y=\"0\"></use><use x=\"1085\" xlink:href=\"#MJMAIN-65\" y=\"0\"></use><use x=\"1529\" xlink:href=\"#MJMAIN-77\" y=\"0\"></use><g transform=\"translate(2252,-150)\"><use transform=\"scale(0.707)\" x=\"0\" xlink:href=\"#MJMAIN-32\" y=\"0\"></use><use transform=\"scale(0.707)\" x=\"500\" xlink:href=\"#MJMATHI-6E\" y=\"0\"></use></g></g><use x=\"3520\" xlink:href=\"#MJMAIN-2C\" y=\"0\"></use><g transform=\"translate(3965,0)\"><use x=\"0\" xlink:href=\"#MJMAIN-50\" y=\"0\"></use><use x=\"681\" xlink:href=\"#MJMAIN-66\" y=\"0\"></use><use x=\"1054\" xlink:href=\"#MJMAIN-66\" y=\"0\"></use></g><use x=\"5391\" xlink:href=\"#MJMAIN-29\" y=\"0\"></use></g></svg><span role=\"presentation\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mo stretchy=\"false\">(</mo><msub><mrow><mi mathvariant=\"normal\">S</mi><mi mathvariant=\"normal\">k</mi><mi mathvariant=\"normal\">e</mi><mi mathvariant=\"normal\">w</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msub><mo>,</mo><mrow><mi mathvariant=\"normal\">P</mi><mi mathvariant=\"normal\">f</mi><mi mathvariant=\"normal\">f</mi></mrow><mo stretchy=\"false\">)</mo></math></span></span><script type=\"math/tex\">(\\textrm{Skew}_{2n},\\textrm{Pff})</script></span>, and <span><span style=\"\"></span><span data-mathml='&lt;math xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi mathvariant=\"normal\"&gt;S&lt;/mi&gt;&lt;mi mathvariant=\"normal\"&gt;y&lt;/mi&gt;&lt;mi mathvariant=\"normal\"&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo movablelimits=\"true\" form=\"prefix\"&gt;det&lt;/mo&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;/math&gt;' role=\"presentation\" style=\"font-size: 100%; display: inline-block; position: relative;\" tabindex=\"0\"><svg aria-hidden=\"true\" focusable=\"false\" height=\"2.614ex\" role=\"img\" style=\"vertical-align: -0.706ex;\" viewbox=\"0 -821.4 5057.8 1125.3\" width=\"11.747ex\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g fill=\"currentColor\" stroke=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use x=\"0\" xlink:href=\"#MJMAIN-28\" y=\"0\"></use><g transform=\"translate(389,0)\"><use x=\"0\" xlink:href=\"#MJMAIN-53\" y=\"0\"></use><use x=\"556\" xlink:href=\"#MJMAIN-79\" y=\"0\"></use><use x=\"1085\" xlink:href=\"#MJMAIN-6D\" y=\"0\"></use><use transform=\"scale(0.707)\" x=\"2713\" xlink:href=\"#MJMATHI-6E\" y=\"-340\"></use></g><use x=\"2832\" xlink:href=\"#MJMAIN-2C\" y=\"0\"></use><g transform=\"translate(3277,0)\"><use xlink:href=\"#MJMAIN-64\"></use><use x=\"556\" xlink:href=\"#MJMAIN-65\" y=\"0\"></use><use x=\"1001\" xlink:href=\"#MJMAIN-74\" y=\"0\"></use></g><use x=\"4668\" xlink:href=\"#MJMAIN-29\" y=\"0\"></use></g></svg><span role=\"presentation\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mo stretchy=\"false\">(</mo><msub><mrow><mi mathvariant=\"normal\">S</mi><mi mathvariant=\"normal\">y</mi><mi mathvariant=\"normal\">m</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mo form=\"prefix\" movablelimits=\"true\">det</mo><mo stretchy=\"false\">)</mo></math></span></span><script type=\"math/tex\">(\\textrm{Sym}_{n}, \\det )</script></span>. We then obtain an asymptotic for </p><span><span style=\"\"></span><span data-mathml='&lt;math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"&gt;&lt;msub&gt;&lt;mi&gt;&amp;#x03C0;&lt;/mi&gt;&lt;mrow&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi mathvariant=\"normal\"&gt;&amp;#x0023;&lt;/mi&gt;&lt;mo fence=\"false\" stretchy=\"false\"&gt;{&lt;/mo&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;&amp;#x2208;&lt;/mo&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;mo movablelimits=\"true\" form=\"prefix\"&gt;max&lt;/mo&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;mrow&gt;&lt;mo stretchy=\"false\"&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;msub&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo stretchy=\"false\"&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;mo&gt;&amp;#x2264;&lt;/mo&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;mtext&gt;&amp;#xA0;is prime&lt;/mtext&gt;&lt;mo fence=\"false\" stretchy=\"false\"&gt;}&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;/math&gt;' role=\"presentation\" style=\"font-size: 100%; display: inline-block; position: relative;\" tabindex=\"0\"><svg aria-hidden=\"true\" focusable=\"false\" height=\"2.714ex\" role=\"img\" style=\"vertical-align: -0.806ex;\" viewbox=\"0 -821.4 22388.7 1168.4\" width=\"52ex\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g fill=\"currentColor\" stroke=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use x=\"0\" xlink:href=\"#MJMATHI-3C0\" y=\"0\"></use><g transform=\"translate(570,-150)\"><use transform=\"scale(0.707)\" x=\"0\" xlink:href=\"#MJMATHI-56\" y=\"0\"></use><use transform=\"scale(0.707)\" x=\"769\" xlink:href=\"#MJMAIN-2C\" y=\"0\"></use><use transform=\"scale(0.707)\" x=\"1048\" xlink:href=\"#MJMATHI-46\" y=\"0\"></use></g><use x=\"1941\" xlink:href=\"#MJMAIN-28\" y=\"0\"></use><use x=\"2331\" xlink:href=\"#MJMATHI-54\" y=\"0\"></use><use x=\"3035\" xlink:href=\"#MJMAIN-29\" y=\"0\"></use><use x=\"3702\" xlink:href=\"#MJMAIN-3D\" y=\"0\"></use><use x=\"4759\" xlink:href=\"#MJMAIN-23\" y=\"0\"></use><use x=\"5592\" xlink:href=\"#MJMAIN-7B\" y=\"0\"></use><use x=\"6093\" xlink:href=\"#MJMATHI-76\" y=\"0\"></use><use x=\"6856\" xlink:href=\"#MJMAIN-2208\" y=\"0\"></use><use x=\"7801\" xlink:href=\"#MJMATHI-56\" y=\"0\"></use><use x=\"8848\" xlink:href=\"#MJMAIN-3A\" y=\"0\"></use><g transform=\"translate(9405,0)\"><use xlink:href=\"#MJMAIN-6D\"></use><use x=\"833\" xlink:href=\"#MJMAIN-61\" y=\"0\"></use><use x=\"1334\" xlink:href=\"#MJMAIN-78\" y=\"0\"></use></g><use x=\"11267\" xlink:href=\"#MJMAIN-28\" y=\"0\"></use><use x=\"11657\" xlink:href=\"#MJMAIN-7C\" y=\"0\"></use><g transform=\"translate(11935,0)\"><use x=\"0\" xlink:href=\"#MJMATHI-76\" y=\"0\"></use><use transform=\"scale(0.707)\" x=\"686\" xlink:href=\"#MJMATHI-69\" y=\"-213\"></use></g><use x=\"12765\" xlink:href=\"#MJMAIN-7C\" y=\"0\"></use><use x=\"13043\" xlink:href=\"#MJMAIN-29\" y=\"0\"></use><use x=\"13711\" xlink:href=\"#MJMAIN-2264\" y=\"0\"></use><use x=\"14767\" xlink:href=\"#MJMATHI-54\" y=\"0\"></use><use x=\"15472\" xlink:href=\"#MJMAIN-2C\" y=\"0\"></use><use x=\"15917\" xlink:href=\"#MJMATHI-46\" y=\"0\"></use><use x=\"16666\" xlink:href=\"#MJMAIN-28\" y=\"0\"></use><use x=\"17056\" xlink:href=\"#MJMATHI-76\" y=\"0\"></use><use x=\"17541\" xlink:href=\"#MJMAIN-29\" y=\"0\"></use><g transform=\"translate(17931,0)\"><use x=\"250\" xlink:href=\"#MJMAIN-69\" y=\"0\"></use><use x=\"528\" xlink:href=\"#MJMAIN-73\" y=\"0\"></use><use x=\"1173\" xlink:href=\"#MJMAIN-70\" y=\"0\"></use><use x=\"1729\" xlink:href=\"#MJMAIN-72\" y=\"0\"></use><use x=\"2122\" xlink:href=\"#MJMAIN-69\" y=\"0\"></use><use x=\"2400\" xlink:href=\"#MJMAIN-6D\" y=\"0\"></use><use x=\"3234\" xlink:href=\"#MJMAIN-65\" y=\"0\"></use></g><use x=\"21609\" xlink:href=\"#MJMAIN-7D\" y=\"0\"></use><use x=\"22110\" xlink:href=\"#MJMAIN-2C\" y=\"0\"></use></g></svg><span role=\"presentation\"><math display=\"block\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mi>π</mi><mrow><mi>V</mi><mo>,</mo><mi>F</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>T</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mi mathvariant=\"normal\">#</mi><mo fence=\"false\" stretchy=\"false\">{</mo><mi>v</mi><mo>∈</mo><mi>V</mi><mo>:</mo><mo form=\"prefix\" movablelimits=\"true\">max</mo><mo stretchy=\"false\">(</mo><mrow><mo stretchy=\"false\">|</mo></mrow><msub><mi>v</mi><mrow><mi>i</mi></mrow></msub><mrow><mo stretchy=\"false\">|</mo></mrow><mo stretchy=\"false\">)</mo><mo>≤</mo><mi>T</mi><mo>,</mo><mi>F</mi><mo stretchy=\"false\">(</mo><mi>v</mi><mo stretchy=\"false\">)</mo><mtext> is prime</mtext><mo fence=\"false\" stretchy=\"false\">}</mo><mo>,</mo></math></span></span><script type=\"math/tex; mode=display\"> \\pi _{V,F}(T)= \\#\\{v\\in V: \\max (|v_{i}|)\\leq T, F(v) \\text{ is prime}\\}, </script></span><p> that matches the Bateman-Horn prediction.</p><p>The key ingredients of our proof are an asymptotic count for integral points on the level sets of <span><span style=\"\"></span><span data-mathml='&lt;math xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/math&gt;' role=\"presentation\" style=\"font-size: 100%; display: inline-block; position: relative;\" tabindex=\"0\"><svg aria-hidden=\"true\" focusable=\"false\" height=\"1.909ex\" role=\"img\" style=\"vertical-align: -0.205ex;\" viewbox=\"0 -733.9 749.5 822.1\" width=\"1.741ex\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g fill=\"currentColor\" stroke=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use x=\"0\" xlink:href=\"#MJMATHI-46\" y=\"0\"></use></g></svg><span role=\"presentation\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>F</mi></math></span></span><script type=\"math/tex\">F</script></span> given by Linnik equidistribution, a geometric approximation of the box by cones, and an upper bound sieve to bound the number of prime values missed by the approximation. In the case of the determinant polynomial on symmetric matrices, we must also use the Siegel mass formula to compute the product of local densities for the main term.</p>","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":"31 1","pages":""},"PeriodicalIF":2.5000,"publicationDate":"2025-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Geometric and Functional Analysis","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00039-025-00716-y","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0

Abstract

We establish a new class of examples of the multivariate Bateman-Horn conjecture by using tools from dynamics. These cases include the determinant polynomial on the space of n×n matrices, the Pfaffian on the space of skew-symmetric 2n×2n matrices, and the determinant polynomial on the space of symmetric n×n matrices. In particular, let (V,F) be any pair among the following: (Matn,det), (Skew2n,Pff), and (Symn,det). We then obtain an asymptotic for

πV,F(T)=#{vV:max(|vi|)T,F(v) is prime},

that matches the Bateman-Horn prediction.

The key ingredients of our proof are an asymptotic count for integral points on the level sets of F given by Linnik equidistribution, a geometric approximation of the box by cones, and an upper bound sieve to bound the number of prime values missed by the approximation. In the case of the determinant polynomial on symmetric matrices, we must also use the Siegel mass formula to compute the product of local densities for the main term.

齐次动力学多项式的素数定理
利用动力学工具建立了一类新的多元Bateman-Horn猜想的实例。这些情况包括n×nn \times n个矩阵空间上的行列式多项式,偏对称2n×2n2n \times 2n个矩阵空间上的Pfaffian,以及对称n×nn \times n个矩阵空间上的行列式多项式。特别地,设(V,F)(V,F)是下列任意对:(Matn,det)(\textrm{Mat_n}, {}\det), (Skew2n,Pff)(\textrm{Skew_2n},{}\textrm{Pff})和(Symn,det)(\textrm{Sym_n}, {}\det)。然后,我们得到了πV,F(T)=#{v∈v:max(|vi|)≤T,F(v)是质数},\pi _V,F{(T)= #{v }\in v: \max (|{v_i}|) \leq T,F(v) \text{ is prime}}的渐近,符合batemanan - horn预测。我们的证明的关键成分是由Linnik等分布给出的FF水平集上的积分点的渐近计数,由锥体给出的盒子的几何近似,以及一个上界筛来限制近似错过的素数值的数量。在对称矩阵上的行列式多项式的情况下,我们还必须使用西格尔质量公式来计算主项的局部密度积。
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来源期刊
CiteScore
3.70
自引率
4.50%
发文量
34
审稿时长
6-12 weeks
期刊介绍: Geometric And Functional Analysis (GAFA) publishes original research papers of the highest quality on a broad range of mathematical topics related to geometry and analysis. GAFA scored in Scopus as best journal in "Geometry and Topology" since 2014 and as best journal in "Analysis" since 2016. Publishes major results on topics in geometry and analysis. Features papers which make connections between relevant fields and their applications to other areas.
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