{"title":"多项式无幂值密度 II","authors":"Kostadinka Lapkova , Stanley Yao Xiao","doi":"10.1016/j.jnt.2024.06.010","DOIUrl":null,"url":null,"abstract":"<div><p>In this paper we prove that polynomials <span><math><mi>F</mi><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo><mo>∈</mo><mi>Z</mi><mo>[</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>]</mo></math></span> of degree <span><math><mi>d</mi><mo>≥</mo><mn>3</mn></math></span>, satisfying certain hypotheses, take on the expected density of <span><math><mo>(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span>-free values. This extends the authors' earlier result in <span><span>[14]</span></span> where a different method implied the similar statement for polynomials of degree <span><math><mi>d</mi><mo>≥</mo><mn>5</mn></math></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/S0022314X24001550/pdfft?md5=5679964f477441d43dd0509c9504b52e&pid=1-s2.0-S0022314X24001550-main.pdf","citationCount":"0","resultStr":"{\"title\":\"Density of power-free values of polynomials II\",\"authors\":\"Kostadinka Lapkova , Stanley Yao Xiao\",\"doi\":\"10.1016/j.jnt.2024.06.010\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>In this paper we prove that polynomials <span><math><mi>F</mi><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo><mo>∈</mo><mi>Z</mi><mo>[</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>]</mo></math></span> of degree <span><math><mi>d</mi><mo>≥</mo><mn>3</mn></math></span>, satisfying certain hypotheses, take on the expected density of <span><math><mo>(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span>-free values. This extends the authors' earlier result in <span><span>[14]</span></span> where a different method implied the similar statement for polynomials of degree <span><math><mi>d</mi><mo>≥</mo><mn>5</mn></math></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/S0022314X24001550/pdfft?md5=5679964f477441d43dd0509c9504b52e&pid=1-s2.0-S0022314X24001550-main.pdf\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0022314X24001550\",\"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/S0022314X24001550","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
In this paper we prove that polynomials of degree , satisfying certain hypotheses, take on the expected density of -free values. This extends the authors' earlier result in [14] where a different method implied the similar statement for polynomials of degree .
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