Pure and Applied Mathematics Quarterly
Volume 12 (2016)
On the sharp polynomial upper estimate conjecture in eight-dimensional simplex
Pages: 353 – 398
The estimate of integral points in an $n$-dimensional polyhedron has many applications in singularity theory, number theory and toric geometry. The third author formulated Yau Number Theoretic Conjecture which gives a sharp polynomial upper estimate on the number of positive integral points in $n$-dimensional $(n \geq 3)$ real right-angled simplices. The previous results on the conjecture in low dimension cases $(n \leq 6)$ have been proved by using the sharp GLY conjecture. However, it is only valid up to $6$ dimension. The Yau Number Theoretic Conjecture for $n = 7$ has been shown with a completely new method in “Biggest sharp polynomial estimate of integral points in right-angled simplices” [S. S.-T. Yau, B. Yuan and H. Q. Zuo. J. Number Theory, 160 (2016), 254–286]. In this paper, on the one hand, we use similar method to prove the conjecture for $n = 8$ case, but with more meticulous analyses. The main method of proof is summing existing sharp upper bounds for the number of points in seven-dimensional simplex over the cross sections of eight-dimensional simplex. This is a significant progress since it sheds light on proving the Yau Number Theoretic Conjecture in general case. On the other hand, we give a new sharper estimate of the Dickman-De Bruijn function $\psi (x, y)$ for $5 \leq y \lt 23$, compared with the result obtained by Ennola [“On numbers with small prime divisors”. Ann. Acad. Sci. Fenn. Ser. Al. 440 (1969), 1–16].
sharp upper estimate, integral points, prime decomposition, simplex
2010 Mathematics Subject Classification
Primary 52B20. Secondary 11P21, 32S05.
This work was partially supported by NSCF (grant nos. 11531007, 11771231, 11401335), Tsinghua University Initiative Scientific Research Program and Ministry of Science and Technology R.O.C. and Chang Gung Memorial Hospital (grant no. MOST 106-2115-M-255-001, NMRPF3G0101).
Received 12 December 2017
Published 9 February 2018