A Sharp Upper Estimate of the Number of Integral Points in a 5-Dimensional Tetrahedra

Lin, Ke-Pao; Yau, Stephen S.-T.

doi:10.1006/jnth.2001.2720

Cited by 12 publications

(18 citation statements)

References 14 publications

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“…The Yau geometric conjecture was answered affirmatively for n = 3, 4, 5 by [3,22,37], respectively. In order to overcome the difficulty that the GLY sharp estimate conjecture is only true if a n is larger than y(n), a positive integer depending on n, Yau proposes to prove a new sharp polynomial estimate conjecture which is motivated from the Yau geometric conjecture.…”

Section: Conjecture 12 (Yau Geometric Conjecture)mentioning

confidence: 94%

“…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. It has also been proven generally for n 6.…”

Section: Introductionmentioning

confidence: 96%

“…Starting from early 1990's, the authors of [22,36,38] tried to get sharp upper estimates of P n where a i 's are positive real numbers. They were successful for n = 3, 4 and 5.…”

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%

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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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Section: Conjecture 12 (Yau Geometric Conjecture)mentioning

confidence: 94%

Section: Introductionmentioning

confidence: 96%

“…Starting from early 1990's, the authors of [22,36,38] tried to get sharp upper estimates of P n where a i 's are positive real numbers. They were successful for n = 3, 4 and 5.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

2015

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“…Starting from early 90's, Yau, Xu and Lin [22,37,39] tried to get a sharp upper estimates of P n when a i are just positive real numbers. They were able to obtain it under certain conditions, specifically when n = 3, 4, and 5.…”

Section: Introductionmentioning

confidence: 99%

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 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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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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A Sharp Upper Estimate of the Number of Integral Points in a 5-Dimensional Tetrahedra

Cited by 12 publications

References 14 publications

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

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

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

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