{"title":"多项式积的贫乏现象","authors":"Victor Y. Wang, Max Wenqiang Xu","doi":"10.1112/blms.13095","DOIUrl":null,"url":null,"abstract":"<p>Let <span></span><math>\n <semantics>\n <mrow>\n <mi>P</mi>\n <mo>(</mo>\n <mi>x</mi>\n <mo>)</mo>\n <mo>∈</mo>\n <mi>Z</mi>\n <mo>[</mo>\n <mi>x</mi>\n <mo>]</mo>\n </mrow>\n <annotation>$P(x)\\in \\mathbb {Z}[x]$</annotation>\n </semantics></math> be a polynomial with at least two distinct complex roots. We prove that the number of solutions <span></span><math>\n <semantics>\n <mrow>\n <mrow>\n <mo>(</mo>\n <msub>\n <mi>x</mi>\n <mn>1</mn>\n </msub>\n <mo>,</mo>\n <mi>⋯</mi>\n <mo>,</mo>\n <msub>\n <mi>x</mi>\n <mi>k</mi>\n </msub>\n <mo>,</mo>\n <msub>\n <mi>y</mi>\n <mn>1</mn>\n </msub>\n <mo>,</mo>\n <mi>⋯</mi>\n <mo>,</mo>\n <msub>\n <mi>y</mi>\n <mi>k</mi>\n </msub>\n <mo>)</mo>\n </mrow>\n <mo>∈</mo>\n <msup>\n <mrow>\n <mo>[</mo>\n <mi>N</mi>\n <mo>]</mo>\n </mrow>\n <mrow>\n <mn>2</mn>\n <mi>k</mi>\n </mrow>\n </msup>\n </mrow>\n <annotation>$(x_1, \\dots, x_k, y_1, \\dots, y_k)\\in [N]^{2k}$</annotation>\n </semantics></math> to the equation\n\n </p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"56 8","pages":"2718-2726"},"PeriodicalIF":0.8000,"publicationDate":"2024-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Paucity phenomena for polynomial products\",\"authors\":\"Victor Y. Wang, Max Wenqiang Xu\",\"doi\":\"10.1112/blms.13095\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Let <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>P</mi>\\n <mo>(</mo>\\n <mi>x</mi>\\n <mo>)</mo>\\n <mo>∈</mo>\\n <mi>Z</mi>\\n <mo>[</mo>\\n <mi>x</mi>\\n <mo>]</mo>\\n </mrow>\\n <annotation>$P(x)\\\\in \\\\mathbb {Z}[x]$</annotation>\\n </semantics></math> be a polynomial with at least two distinct complex roots. We prove that the number of solutions <span></span><math>\\n <semantics>\\n <mrow>\\n <mrow>\\n <mo>(</mo>\\n <msub>\\n <mi>x</mi>\\n <mn>1</mn>\\n </msub>\\n <mo>,</mo>\\n <mi>⋯</mi>\\n <mo>,</mo>\\n <msub>\\n <mi>x</mi>\\n <mi>k</mi>\\n </msub>\\n <mo>,</mo>\\n <msub>\\n <mi>y</mi>\\n <mn>1</mn>\\n </msub>\\n <mo>,</mo>\\n <mi>⋯</mi>\\n <mo>,</mo>\\n <msub>\\n <mi>y</mi>\\n <mi>k</mi>\\n </msub>\\n <mo>)</mo>\\n </mrow>\\n <mo>∈</mo>\\n <msup>\\n <mrow>\\n <mo>[</mo>\\n <mi>N</mi>\\n <mo>]</mo>\\n </mrow>\\n <mrow>\\n <mn>2</mn>\\n <mi>k</mi>\\n </mrow>\\n </msup>\\n </mrow>\\n <annotation>$(x_1, \\\\dots, x_k, y_1, \\\\dots, y_k)\\\\in [N]^{2k}$</annotation>\\n </semantics></math> to the equation\\n\\n </p>\",\"PeriodicalId\":55298,\"journal\":{\"name\":\"Bulletin of the London Mathematical Society\",\"volume\":\"56 8\",\"pages\":\"2718-2726\"},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2024-05-31\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Bulletin of the London Mathematical Society\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1112/blms.13095\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Bulletin of the London Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/blms.13095","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
让 P ( x ) ∈ Z [ x ] $P(x)\in \mathbb {Z}[x]$ 是一个至少有两个不同复根的多项式。我们证明解的个数 ( x 1 , ⋯ , x k , y 1 , ⋯ , y k ) ∈[N]2k$(x_1, \dots, x_k, y_1, \dots, y_k)\in [N]^{2k}$ 解方程
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