Double Gauss Sums

双高斯和

• 批准号: 418029-2013
• 负责人: Alaca, Ayse
• 金额: $ 1.09万
• 依托单位: Carleton University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=696265
• 关键词: Double; Gauss; Sums

Gauss sums are fundamental objects in number theory and arithmetic geometry. A number of generalizations of Gauss sums appear in the literature. They have extensive applications in many areas of mathematics and computer science. I plan to study the two dimensional quadratic Gauss sums (double Gauss sums) and their applications. My main objective is to evaluate the double Gauss sums explicitly. I aim to use their evaluations to determine a general formula for the number of solutions of certain quadratic equations over residue rings modulo prime powers. As a long term objective, upon completing the evaluations of double Gauss sums, I plan to study multi-dimensional quadratic Gauss sums, and apply them to find the number of solutions of corresponding quadratic equations over residue rings. I also plan to use my research results on double Gauss sums together with Siegel's mass formula (the local density approach) to find the number of representations of positive integers by quadratic forms with integer coefficients.

高斯和是数论和算术几何中的基本对象。文献中出现了许多高斯和的推广。它们在数学和计算机科学的许多领域都有广泛的应用。