数论和p-adic分析

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.

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.

利用Igusa稳定相位公式和牛顿多面体理论,证明了三元Hauser混合多项式的Igusa局部zeta函数是q^s的有理函数,并给出了它的所有候选极点.利用p-adic方法和解析方法,证明了当n>1时，单位分数1/a^{s_1}, 1/(a+b)^{s_2},...,1/(a+(n-1)b)^{s_n}的和不是整数,这推广了Nagell定理.进而证明了非负整系数多项式的多重倒数和与多重倒数星和几乎都不是整数.利用p-adic分析和组合方法, 证明了三次级数{n^3+2}连续项的最小公倍数所关联的算术函数为周期算术函数, 且确定了其最小正周期. 利用三次Gauss和以及Davenport-Hasse关系式,证明了有限域上两类三次对角超曲面的有理点个数所对应的生成函数是有理函数,由此解决了关于模p三次同余方程解的个数的Gauss符号问题.利用p-adic Gamma函数和Gross-Koblitz公式,给出了满列秩的Laurent多项式的扭曲指数和的L-函数的明确表达式,确定了该L-函数的p-adic牛顿多边形.本项目所取得的成果具有较重要的理论意义和科学价值,并将对有限域上的算术几何和数论领域的进一步发展产生积极影响.

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