利用p-adic分析中的理论和方法,并利用Igusa关于π-adic积分的稳定相位公式,研究Hauser混合多项式的局部zeta函数的有理性及其极点. 发展p-adic方法,并利用Dwork同余式定理, 研究有理参数和二次参数的超几何级数和超几何镜面映射的N整性和p-adic整性. 利用p-adic赋值和黎曼zeta函数的特殊值, 研究整系数多项式倒数的对称函数的整性. 利用p-adic方法和解析方法,研究三次及更高次级数连续项的最小公倍数的下界,渐近公式和与三次及更高次级数连续项的最小公倍数相关联的算术函数的周期性及最小正周期. 本项目具有较重要的科学理论意义和价值,而且也将会在数论和代数几何中产生积极影响.

By using the theory and approach in p-adic analysis, and using Igusa's stationary phase formula, we study the Igusa's local zeta functions of Hauser hybrid polynomials with coefficients in a non-archimedean local field of positive characteristic. By the Dwork formal congruence and the p-adic method, we study the N-integrality of hypergeometric series with parameters from the rational field and a quadratic field. Using p-adic method, we study the periodicity of the arithmetic function associated with the least common multiple of consecutive cubic progression terms and consecutive high progression terms and determine their smallest periods. Making use of p-adic valuation and the special value of Riemann zeta function, we investigate the integrality of the the elementary symmetric functions of a reciprocal integer polynomial sequence. By p-adic and analytic methods, we investigate the asymptotic behavior of the least common multiple of consecutive cubic progression terms and consecutive high progression terms. This project will be important in science, and will make active and important contributions to number theory and algebraic geometry.