An upper estimate of integral points in real simplices with an application to singularity theory

Yau, Stephen T.; Zhang, Letian

doi:10.4310/mrl.2006.v13.n6.a6

Cited by 22 publications

(21 citation statements)

References 14 publications

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“…The weak version was finally proved by Yau and Zhang [20]. In the same paper, the authors claim the strong version has been checked computationally up to n ≤ 10.…”

Section: A First Application: a Bound "á La Wilf "mentioning

confidence: 90%

Characterization of Gaps and Elements of a Numerical Semigroup Using Groebner Bases

Márquez-Campos¹,

Tornero²

2015

Contemporary Mathematics

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“…The weak version was finally proved by Yau and Zhang [20]. In the same paper, the authors claim the strong version has been checked computationally up to n ≤ 10.…”

Section: A First Application: a Bound "á La Wilf "mentioning

confidence: 90%

Characterization of Gaps and Elements of a Numerical Semigroup Using Groebner Bases

Márquez-Campos¹,

Tornero²

2015

Contemporary Mathematics

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“…They are sharp because the equality holds true if and only if all

a_{i}

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

3 \leq n \leq 5

.…”

Section: Introductionmentioning

confidence: 90%

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 GLY conjecture was proven by Xu and Yau for n = 3 [20] and n = 4 [22], for n = 5, see [4], [6] and [8], Wang and Yau for 3 ≤ n ≤ 6 [18], where β(n) = n − 1 for 3 ≤ n ≤ 6. The rough GLY upper estimate for all n was proven in [24].…”

Section: Where Equality Holds If and Only Ifmentioning

confidence: 97%

“…The GLY conjecture was the first major step towards proving the following conjecture made by Yau in 1995 [24]:…”

Section: Theorem 13 (Lu-ya-zu [11]mentioning

confidence: 99%

Coordinate-free characterization of homogeneous polynomials with isolated singularities

Chen

Lin

Yau

et al. 2011

Communications in Analysis and Geometry

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The Durfee conjecture, proposed in 1978, relates two important invariants of isolated hypersurface singularities by a famous inequality; however, the inequality in this conjecture is not sharp. In 1995, Yau announced his conjecture which proposed a sharp inequality. The Yau conjecture characterizes the conditions under which an affine hypersurface with an isolated singularity at the origin is a cone over a nonsingular projective hypersurface; in other words, the conjecture gives a coordinate-free characterization of when a convergent power series is a homogeneous polynomial after a biholomorphic change of variables. In this paper, we have proved that if p g > 0, then 54) and p g , μ and v are, respectively, the geometric genus, the Milnor number, and the multiplicity of the isolated singularity at the origin of a weighted homogeneous polynomial. As a consequence, we prove that the Yau conjecture holds for n = 5 if p g > 0. In the process, we have also defined yet another sharp upper bound for the number of positive integral points within a five-dimensional simplex.

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An upper estimate of integral points in real simplices with an application to singularity theory

Cited by 22 publications

References 14 publications

Characterization of Gaps and Elements of a Numerical Semigroup Using Groebner Bases

Characterization of Gaps and Elements of a Numerical Semigroup Using Groebner Bases

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

Coordinate-free characterization of homogeneous polynomials with isolated singularities

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