{"title":"关于正向轨道上 (D,S) 积分点的非扎里斯基密度和子空间定理","authors":"Nathan Grieve , Chatchai Noytaptim","doi":"10.1016/j.jnt.2024.06.005","DOIUrl":null,"url":null,"abstract":"<div><p>Working over a base number field <strong>K</strong>, we study the attractive question of Zariski non-density for <span><math><mo>(</mo><mi>D</mi><mo>,</mo><mi>S</mi><mo>)</mo></math></span>-integral points in <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo></math></span> the forward <em>f</em>-orbit of a rational point <span><math><mi>x</mi><mo>∈</mo><mi>X</mi><mo>(</mo><mi>K</mi><mo>)</mo></math></span>. Here, <span><math><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>X</mi></math></span> is a regular surjective self-map for <em>X</em> a geometrically irreducible projective variety over <strong>K</strong>. Given a non-zero and effective <em>f</em>-quasi-polarizable Cartier divisor <em>D</em> on <em>X</em> and defined over <strong>K</strong>, our main result gives a sufficient condition, that is formulated in terms of the <em>f</em>-dynamics of <em>D</em>, for non-Zariski density of certain dynamically defined subsets of <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo></math></span>. For the case of <span><math><mo>(</mo><mi>D</mi><mo>,</mo><mi>S</mi><mo>)</mo></math></span>-integral points, this result gives a sufficient condition for non-Zariski density of integral points in <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo></math></span>. Our approach expands on that of Yasufuku, <span><span>[13]</span></span>, building on earlier work of Silverman <span><span>[11]</span></span>. Our main result gives an unconditional form of the main results of <span><span>[13]</span></span>; the key arithmetic input to our main theorem is the Subspace Theorem of Schmidt in the generalized form that has been given by Ru and Vojta in <span><span>[10]</span></span> and expanded upon in <span><span>[3]</span></span> and <span><span>[6]</span></span>.</p></div>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0022314X24001495/pdfft?md5=b3dd7c5b16ab793f55d50824e16a3394&pid=1-s2.0-S0022314X24001495-main.pdf","citationCount":"0","resultStr":"{\"title\":\"On non-Zariski density of (D,S)-integral points in forward orbits and the Subspace Theorem\",\"authors\":\"Nathan Grieve , Chatchai Noytaptim\",\"doi\":\"10.1016/j.jnt.2024.06.005\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Working over a base number field <strong>K</strong>, we study the attractive question of Zariski non-density for <span><math><mo>(</mo><mi>D</mi><mo>,</mo><mi>S</mi><mo>)</mo></math></span>-integral points in <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo></math></span> the forward <em>f</em>-orbit of a rational point <span><math><mi>x</mi><mo>∈</mo><mi>X</mi><mo>(</mo><mi>K</mi><mo>)</mo></math></span>. Here, <span><math><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>X</mi></math></span> is a regular surjective self-map for <em>X</em> a geometrically irreducible projective variety over <strong>K</strong>. Given a non-zero and effective <em>f</em>-quasi-polarizable Cartier divisor <em>D</em> on <em>X</em> and defined over <strong>K</strong>, our main result gives a sufficient condition, that is formulated in terms of the <em>f</em>-dynamics of <em>D</em>, for non-Zariski density of certain dynamically defined subsets of <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo></math></span>. For the case of <span><math><mo>(</mo><mi>D</mi><mo>,</mo><mi>S</mi><mo>)</mo></math></span>-integral points, this result gives a sufficient condition for non-Zariski density of integral points in <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo></math></span>. Our approach expands on that of Yasufuku, <span><span>[13]</span></span>, building on earlier work of Silverman <span><span>[11]</span></span>. Our main result gives an unconditional form of the main results of <span><span>[13]</span></span>; the key arithmetic input to our main theorem is the Subspace Theorem of Schmidt in the generalized form that has been given by Ru and Vojta in <span><span>[10]</span></span> and expanded upon in <span><span>[3]</span></span> and <span><span>[6]</span></span>.</p></div>\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2024-07-17\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://www.sciencedirect.com/science/article/pii/S0022314X24001495/pdfft?md5=b3dd7c5b16ab793f55d50824e16a3394&pid=1-s2.0-S0022314X24001495-main.pdf\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0022314X24001495\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0022314X24001495","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
On non-Zariski density of (D,S)-integral points in forward orbits and the Subspace Theorem
Working over a base number field K, we study the attractive question of Zariski non-density for -integral points in the forward f-orbit of a rational point . Here, is a regular surjective self-map for X a geometrically irreducible projective variety over K. Given a non-zero and effective f-quasi-polarizable Cartier divisor D on X and defined over K, our main result gives a sufficient condition, that is formulated in terms of the f-dynamics of D, for non-Zariski density of certain dynamically defined subsets of . For the case of -integral points, this result gives a sufficient condition for non-Zariski density of integral points in . Our approach expands on that of Yasufuku, [13], building on earlier work of Silverman [11]. Our main result gives an unconditional form of the main results of [13]; the key arithmetic input to our main theorem is the Subspace Theorem of Schmidt in the generalized form that has been given by Ru and Vojta in [10] and expanded upon in [3] and [6].