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

Abstract

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.

Sharp upper estimate of geometric genus and coordinate-free characterization of isolated homogeneous hypersurface singularities

“…Proof. Before we begin the proof, it will be beneficial to explicitly write out the sharp GLY Conjecture for n = 6, which was proven in [17]. Equality holds if and only if a 1 = a 2 = a 3 = a 4 = a 5 = a 6 ∈ Z + .…”

“…The sharp GLY Conjecture has been proven to be true for 3 ≤ n ≤ 6 [19,4,17,7]. The rough GLY upper estimate for all n was proven by Yau and Zhang [20].…”

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

“…Case (6b) a 6 1 9 (31 + √ 574) ≈ 6.10648. In order to solve ths case we will need to use the sharp estimate of the GLY conjecture for n = 6, which has already been proven by Wang and Yau [35].…”

“…It has also been proven generally for n 6. However, for n = 7, a counterexample to the conjecture has been given in [35]. In [40], a revised verision of GLY conjecture, i.e., Yau-Zhao-Zuo (YZZ) conjecture was proposed and proved to be ture in low dimensions.…”

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

“…The hunt for a good, simple estimate of q(α 1 , ..., α n ) and p(α 1 , ..., α n ) led to several results [7,8,9,15,17,18,19], finally put together in the GLY Conjeture, named after its authors Granville, Lin and Yau.…”