The Sharp Polynomial Upper Estimate In Yau Number Theoretic Conjecture Is Available In7-dimensional Case

Let Pnbe the number of positive integral points in n-dimensional simplex with realedge. For n≥3, the exact form of Pnis rather complicated and hard to figure out howlarge it is. Because its importance in number theory and singularity theory, the problemof giving an polynomial sharp upper estimate of Pnreceives attention by lot of mathe-maticians. In number theory, Pnis related to the Dickman-De Bruijn function, whichcan be used to test the gap between prime numbers. In singularity theory, Pnis closelyrelated to the geometric genus pg. By given a polynomial upper estimate of Pn, Durfeeconjecture was proved. Another problem is, whether we can find an intrinsic charateristicfor a holomorphic function with an isolated singularity at the origin to be a homogeneouspolynomial. To solve this problem, Stephen S.-T. Yau gave the Yau geometric conjecture,which refers to an inequality of pg. To prove the Yau geometric conjecture, Lin, Yau andGranville gave the GLY sharp estimate. However, this estimate has an counter-examplewhen n=7. Although later, this estimate has been improved, the application conditionbecame more strict.So Stephen S.-T Yau gave the Yau Theoretic Conjecture, which givesa sharp polynomial sharp estimate of Pn. The application condition of this estimate doesnot depend on the dimension n. In case the dimension n≤6, this estimate has beenproven available. In this paper, we showed that the estimate is available in case n=7by induction. Our method based on the following points: first, we find a way to judgewhether an one-variable polynomial is nonnegative on some secion; second, by exam-ining its partials and initial values, we transform the problem of judging the sign of themultiple-variable polynomial into several problems of that of one-variable polynomial;third, we divide the problem into several cases such that in each case, the function be-comes a polynomial; last, we make use of the charasteristic of the inequility to save thelabor of computing. What’s more, some of the results in this paper can be applied to thegeneral n case of Yau Theoretic Conjecture.