{"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='<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>n</mi><mo>&#x00D7;</mo><mi>n</mi></math>' 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='<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>2</mn><mi>n</mi><mo>&#x00D7;</mo><mn>2</mn><mi>n</mi></math>' 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='<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>n</mi><mo>&#x00D7;</mo><mi>n</mi></math>' 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='<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>' 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='<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 movablelimits=\"true\" form=\"prefix\">det</mo><mo stretchy=\"false\">)</mo></math>' 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='<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>' 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='<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 movablelimits=\"true\" form=\"prefix\">det</mo><mo stretchy=\"false\">)</mo></math>' 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='<math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"><msub><mi>&#x03C0;</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\">&#x0023;</mi><mo fence=\"false\" stretchy=\"false\">{</mo><mi>v</mi><mo>&#x2208;</mo><mi>V</mi><mo>:</mo><mo movablelimits=\"true\" form=\"prefix\">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>&#x2264;</mo><mi>T</mi><mo>,</mo><mi>F</mi><mo stretchy=\"false\">(</mo><mi>v</mi><mo stretchy=\"false\">)</mo><mtext>&#xA0;is prime</mtext><mo fence=\"false\" stretchy=\"false\">}</mo><mo>,</mo></math>' 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='<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>F</mi></math>' 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 matrices, the Pfaffian on the space of skew-symmetric matrices, and the determinant polynomial on the space of symmetric matrices. In particular, let be any pair among the following: , , and . We then obtain an asymptotic for
that matches the Bateman-Horn prediction.
The key ingredients of our proof are an asymptotic count for integral points on the level sets of 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.
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