Extra Form
Lecturer David Leep
Dept. Univ. of Kentucky
date Mar 17, 2011

It is usually a difficult problem to characterize precisely which elements of a given integral domain can be written as a sum of squares of elements from the integral domain. Let R denote the ring of integers in a quadratic number field. This talk will deal with the problem of identifying which elements of R can be written as a sum of squares. If an element in R can be written as a sum of squares, then the element must be totally positive. This necessary condition is not always sufficient. We will determine exactly when this necessary condition is sufficient. In addition, we will develop several criteria to guarantee that a representation as a sum of squares is possible. The results are based on theorems of I. Niven and C. Siegel from the 1940's, and R. Scharlau from 1980.

Attachment '1'
  1. A new view of Fokker-Planck equations in finite and Infinite dimensional spaces

  2. 원의 유리매개화에 관련된 수학

  3. Introduction to Non-Positively Curved Groups

  4. Noncommutative Geometry. Quantum Space-Time and Diffeomorphism Invariant Geometry

  5. 행렬함수 Permanent의 극소값 결정과 미해결 문제들

  6. The Mathematics of the Bose Gas and its Condensation

  7. Codimension Three Conjecture

  8. 학부생을 위한 강연: 건축과 수학

  9. Classical and Quantum Probability Theory

  10. Iwasawa main conjecture and p-adic L-functions

  11. 학부생을 위한 강연: Choi's orthogonal Latin Squares is at least 61 years earlier than Euler's

  12. 젊은과학자상 수상기념강연: From particle to kinetic and hydrodynamic descriptions to flocking and synchronization

  13. 07Nov
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    Sums of squares in quadratic number rings

  14. Fano manifolds of Calabi-Yau Type

  15. 곡선의 정의란 무엇인가?

  16. The significance of dimensions in mathematics

  17. Fermat´s last theorem

  18. It all started with Moser

  19. On some nonlinear elliptic problems

  20. Topology and number theory

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