Analysis for a sharp polynomial upper estimate of the number of positive integral points in a 4-dimensional tetrahedron

The GLY (Granville-Lin-Yau) Conjecture is a generalization of Lin, Xu and Yau's results. An important application of GLY is its use in characterizing an affine hypersurface in C n as a cone over a nonsingular projective variety. In addition, the Rough Upper Estimate Conjecture in GLY, recently proved by Yau and Zhang, implies the Durfee Conjecture in singularity theory. This paper develops a unified approach to prove the Sharp Upper Estimate Conjecture for general n. Using this unified approach, we prove that the Sharp Upper Estimate Conjecture is true for n = 4, 5, 6. After giving a counter-example to show that the Sharp Upper Estimate Conjecture is not true for n = 7, we propose a Modified GLY Conjecture. For each fixed n, our unified approach can be used to prove this Modified GLY Conjecture.

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“…When n = 3, 4 and 5, this conjecture is true [22,23,37,39]. The sharp estimate conjecture was first formulated in [24].…”

Section: Gly (Granville-linmentioning

confidence: 93%

“…The sharp estimate conjecture was first formulated in [24]. In private communication, to the second author, Granville formulated this sharp estimate conjecture independently after reading [23]. Notice that the sharp estimate conjecture is for n-dimensional real right-angled simplices with a n n − 1.…”

Section: Gly (Granville-linmentioning

confidence: 98%

On the GLY Conjecture of upper estimate of positive integral points in real right-angled simplices

Wang

Yau

2007

Journal of Number Theory

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“…The weak estimate in has recently been proven true by the authors of . Before that, , , , showed that holds for

3 \leq n \leq 5

. The sharp estimate conjecture was first formulated in .…”

Section: Introductionmentioning

confidence: 96%

A sharp polynomial estimate of positive integral points in a 4-dimensional tetrahedron and a sharp estimate of the Dickman-de Bruijn function

2014

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The estimate of integral points in right‐angled simplices has many applications in number theory, complex geometry, toric variety and tropical geometry. In [24], [25], [27], the second author and other coworkers gave a sharp upper estimate that counts the number of positive integral points in n dimensional (n≥3) real right‐angled simplices with vertices whose distance to the origin are at least n−1. A natural problem is how to form a new sharp estimate without the minimal distance assumption. In this paper, we formulate the Number Theoretic Conjecture which is a direct correspondence of the Yau Geometry conjecture. We have proved this conjecture for n=4. This paper gives hope to prove the new conjecture in general. As an application, we give a sharp estimate of the Dickman‐de Bruijn function ψ(x,y) for y<11.

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“…The sharp estimate conjecture was first formulated in [23]. In private communication to the second author, Granville formulated this sharp estimated conjecture independently after reading [21]. Again, the sharp GLY conjecture has been proven individually for n = 3, 4, 5 by [22,37,38], respectively.…”

Section: Introductionmentioning

confidence: 99%

“…They are sharp because the equality holds true if and only if all a i 's take the same integer. In [21,22,36,38], the authors showed that (1.5) holds for 3 n 5. The sharp estimate conjecture was first formulated in [23].…”

Section: Introductionmentioning

confidence: 99%

A sharp estimate of positive integral points in 6-dimensional polyhedra and a sharp estimate of smooth numbers

2015

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Inspired by Durfee Conjecture in singularity theory, Yau formulated the Yau number theoretic conjecture (see Conjecture 1.3) which gives a sharp polynomial upper bound of the number of positive integral points in an n-dimensional (n 3) polyhedron. It is well known that getting the estimate of integral points in the polyhedron is equivalent to getting the estimate of the de Bruijn function ψ(x, y), which is important and has a number of applications to analytic number theory and cryptography. We prove the Yau number theoretic conjecture for n = 6. As an application, we give a sharper estimate of function ψ(x, y) for 5 y < 17, compared with the result obtained by Ennola.

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Analysis for a sharp polynomial upper estimate of the number of positive integral points in a 4-dimensional tetrahedron

Cited by 7 publications

References 6 publications

On the GLY Conjecture of upper estimate of positive integral points in real right-angled simplices

On the GLY Conjecture of upper estimate of positive integral points in real right-angled simplices

A sharp polynomial estimate of positive integral points in a 4-dimensional tetrahedron and a sharp estimate of the Dickman-de Bruijn function

A sharp estimate of positive integral points in 6-dimensional polyhedra and a sharp estimate of smooth numbers

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