Mathematical Sciences: Topics in Arithmetic Theory of Quadratic Forms

数学科学：二次形式算术理论专题

• 批准号: 8803805
• 负责人: John Hsia
• 金额: --
• 依托单位: Ohio State University Research Foundation -DO NOT USE
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1988
• 资助国家: 美国
• 起止时间: 1988-06-01 至 1992-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8803805&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Topics; Arithmetic; Theory

This research concerns the arithmetic theory of quadratic forms. In particular there will be special emphasis on the classification and representation problems of positive definite forms. With respect to the classification theory the Principal Investigator will investigate both the explicit and complete determination of certain genera as well as the relationships between even unimodular quadratic forms over integers of a number field and their algebraic descents. In the area of representation of quadratic forms he will study primitive representations of n-ary forms by positive definite m-ary quadratic forms and representations of binary forms by certain positive quaternary quadratic forms. Quadratic forms are homogeneous polynomials in many variables of degree two. This research concerns the number theory problems which arise in determining which integers can be values of them when integer are substituted for their variables. Classifying quadratic forms in classes that all have the same values and representing the forms by simpler forms are valuable tools in answering this problem.

本研究涉及二次型的算术理论。 特别是，将特别强调正定形式的分类和表示问题。 关于分类理论，首席研究员将研究某些属的显式且完整的确定，以及数域整数上的单模二次形式及其代数下降之间的关系。 在二次形式的表示领域，他将研究正定m元二次形式对n元形式的原始表示以及某些正四元二次形式对二元形式的表示。 二次形式是许多二阶变量的齐次多项式。 这项研究涉及数论问题，这些问题是在用整数代替变量时确定哪些整数可以作为它们的值时出现的。 将二次形式分类为具有相同值的类并用更简单的形式表示形式是回答此问题的宝贵工具。