{"title":"特殊立方零点和对偶变化","authors":"Victor Y. Wang","doi":"10.1112/jlms.12975","DOIUrl":null,"url":null,"abstract":"<p>Let <span></span><math>\n <semantics>\n <mi>F</mi>\n <annotation>$F$</annotation>\n </semantics></math> be a diagonal cubic form over <span></span><math>\n <semantics>\n <mi>Z</mi>\n <annotation>$\\mathbb {Z}$</annotation>\n </semantics></math> in six variables. From the dual variety in the delta method of Duke–Friedlander–Iwaniec and Heath-Brown, we unconditionally extract a weighted count of certain special integral zeros of <span></span><math>\n <semantics>\n <mi>F</mi>\n <annotation>$F$</annotation>\n </semantics></math> in regions of diameter <span></span><math>\n <semantics>\n <mrow>\n <mi>X</mi>\n <mo>→</mo>\n <mi>∞</mi>\n </mrow>\n <annotation>$X \\rightarrow \\infty$</annotation>\n </semantics></math>. Heath-Brown did the same in four variables, but our analysis differs and captures some novel features. We also put forth an axiomatic framework for more general <span></span><math>\n <semantics>\n <mi>F</mi>\n <annotation>$F$</annotation>\n </semantics></math>.</p>","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.12975","citationCount":"0","resultStr":"{\"title\":\"Special cubic zeros and the dual variety\",\"authors\":\"Victor Y. Wang\",\"doi\":\"10.1112/jlms.12975\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Let <span></span><math>\\n <semantics>\\n <mi>F</mi>\\n <annotation>$F$</annotation>\\n </semantics></math> be a diagonal cubic form over <span></span><math>\\n <semantics>\\n <mi>Z</mi>\\n <annotation>$\\\\mathbb {Z}$</annotation>\\n </semantics></math> in six variables. From the dual variety in the delta method of Duke–Friedlander–Iwaniec and Heath-Brown, we unconditionally extract a weighted count of certain special integral zeros of <span></span><math>\\n <semantics>\\n <mi>F</mi>\\n <annotation>$F$</annotation>\\n </semantics></math> in regions of diameter <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>X</mi>\\n <mo>→</mo>\\n <mi>∞</mi>\\n </mrow>\\n <annotation>$X \\\\rightarrow \\\\infty$</annotation>\\n </semantics></math>. Heath-Brown did the same in four variables, but our analysis differs and captures some novel features. We also put forth an axiomatic framework for more general <span></span><math>\\n <semantics>\\n <mi>F</mi>\\n <annotation>$F$</annotation>\\n </semantics></math>.</p>\",\"PeriodicalId\":1,\"journal\":{\"name\":\"Accounts of Chemical Research\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":16.4000,\"publicationDate\":\"2024-08-14\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.12975\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Accounts of Chemical Research\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1112/jlms.12975\",\"RegionNum\":1,\"RegionCategory\":\"化学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"CHEMISTRY, MULTIDISCIPLINARY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/jlms.12975","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
引用次数: 0
摘要
设 F $F$ 是六变量 Z $\mathbb {Z}$ 上的对角立方形式。从杜克-弗里德兰德-伊瓦尼茨（Duke-Friedlander-Iwaniec）和希斯-布朗（Heath-Brown）的三角法中的对偶变化中，我们无条件地提取了直径为 X → ∞ $X \rightarrow \infty$ 的区域中 F $F$ 的某些特殊积分零点的加权计数。希斯-布朗在四个变量中做了同样的工作，但我们的分析有所不同，并捕捉到了一些新的特征。我们还为更一般的 F $F$ 提出了一个公理框架。
Let be a diagonal cubic form over in six variables. From the dual variety in the delta method of Duke–Friedlander–Iwaniec and Heath-Brown, we unconditionally extract a weighted count of certain special integral zeros of in regions of diameter . Heath-Brown did the same in four variables, but our analysis differs and captures some novel features. We also put forth an axiomatic framework for more general .
期刊介绍：
Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance.
Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.
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