The Sharp Algebraic Estimate Of P_n In Yau Number Theoretic Conjecture

Inspired by Durfee conjecture,S.S.-T.Yau established the eponymous number theoretic conjecture and geometric conjecture.LetP_n be the number of positive integral points in n-dimensional simplex with real edge.Yau number theoretic conjecture is mainly to estimateP_n.The sharp polynomial upper estimate ofP_n has been proved when n?7 in Yau number theoretic conjecture,but the lower estimate ofP_n has not been found.This thesis employs and improves the previous method to prove the sharp algebraic lower estimate and the sharp polynomial lower estimate of P₂ by separating a single term from the summation ofP₂.In the process of proving the accuracy of the estimate,this thesis uses the method of constructing the approximation sequence to overcome the difficulty that the inequality cannot be saturated when the lower bound ofP₂ is estimated.This thesis uses the relevant conclusions of the sharp algebraic lower estimate ofP₂ to prove the sharp algebraic lower estimate ofP₃.Similarly to the 2-dimensional case,this thesis proves the accuracy of this estimate ofP₃.In this thesis,a polynomial rough lower estimate of P₃ is given,and a polynomial upper estimate ofP₃ that is sharper than the previous results is given when the edge length meets certain conditions.