On the polynomial sharp upper estimate conjecture in 7-dimensional simplex
| Autor: | Huaiqing Zuo, Beihui Yuan, Stephen S.-T. Yau |
|---|---|
| Rok vydání: | 2016 |
| Předmět: |
Polynomial
Lemma (mathematics) Algebra and Number Theory Conjecture Elliott–Halberstam conjecture 010308 nuclear & particles physics 010102 general mathematics abc conjecture 01 natural sciences Collatz conjecture Combinatorics Number theory 0103 physical sciences 0101 mathematics Lonely runner conjecture Mathematics |
| Zdroj: | Journal of Number Theory. 160:254-286 |
| ISSN: | 0022-314X |
| DOI: | 10.1016/j.jnt.2015.08.012 |
| Popis: | Because of its importance in number theory and singularity theory, the problem of finding a polynomial sharp upper estimate of the number of positive integral points in an n-dimensional ( n ≥ 3 ) polyhedron has received attention by a lot of mathematicians. The first named author proposed the Number Theoretic Conjecture for the upper estimate. The previous results on the Number Theoretic Conjecture in low dimension cases ( n 7 ) are proved by using the sharp GLY conjecture which is true only for low dimensional case. Thus the proof cannot be generalized to high dimension. In this paper, we offer a uniform approach to prove the Number Theoretic Conjecture for all dimensions by simply using the induction method and the Yau–Zhang [19] estimates (see Lemma 2.3 , Lemma 2.4 , Lemma 2.5 ). As a result, the Number Theoretic Conjecture is proven for n = 7 . An important estimate for all dimensions is also obtained ( Proposition 3.1 , Proposition 3.2 ) which will be useful to prove the general case of the Number Theoretic Conjecture. As an application, we give a sharper estimate of the Dickman–De Bruijn function ψ ( x , y ) for 5 ≤ y 19 , compared with the result obtained by Ennola. |
| Databáze: | OpenAIRE |
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