恒隆數學獎

6. 觀塘瑪利諾書院 Investigation on Riemann Zeta Function and P-adic Number In this paper, we will study the connection between the two concepts that are the Riemann zeta function ζ(s) and the p-adic number system. We have found that there is a p-adic analogue of zeta function, ζp(s), and its generalization, the p-adic L-function Lp(s, χ), is very important to modern mathematics such as the studies of elliptic curves and modular forms. 7. 香港培正中學 On Non-Torsion Solutions of Homogeneous Linear Systems over Rings In this paper, we study the existence of non-torsion solutions of a homogeneous linear system over a commutative ring. More precisely, we determine the minimal positive integer n such that any homogeneous systems of m equations with n variables over a given ring R gives a non-torsion solution, i.e. a solution x = (x1,x2,…,xn) such that at least one coordinate xi is not a zero-divisor. We proved that over Noetherian rings, a non-trivial lower bound to the minimal number can be guaranteed via the use of primary decomposition. We also consider the number of generators of ideals in R and the localisations of R. For some classes of Noetherian rings, such as principal ideal rings and reduced rings, we show that such minimal number exists.

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