On a number theoretic conjecture on positive integral points in a 5-dimensional tetrahedron and a sharp estimate of the Dickman–De Bruijn function

Lin, Ke-Pao; Luo, Xue; Yau, Stephen S.-T.; Zuo, Huancong

doi:10.4171/jems/480

Cited by 7 publications

(4 citation statements)

References 25 publications

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“…The purpose of this paper is to prove that the number theoretic conjecture is true for n = 6. The strategy of this paper is different from our previous papers [20,24]. Here for the case n = 6 the techniques used are more complicated than those in the case n 5 and the feasibility of the strategy has been challenged, even if the dimension has only been increased by 1.…”

Section: Conjecture 12 (Yau Geometric Conjecture)mentioning

confidence: 81%

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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: 81%

“…For n = 4, 5, the conjecture has been shown in our previous work [20,24]. The purpose of this paper is to prove that the number theoretic conjecture is true for n = 6.…”

Section: Conjecture 12 (Yau Geometric Conjecture)mentioning

confidence: 84%

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

2015

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“…In recent years, great progress has been made in the calculation of the number of integral points in n-dimensional simplex with non-integral vertice (cf. [1,2]). Such an estimate could be applied to find large gaps between primes, to Waring's problem, to primality testing and factoring algorithms, to singularity theory (cf.…”

Section: Introductionmentioning

confidence: 99%

Proposing a New Theorem to Determine If an Algebraic Polynomial Is Nonnegative in an Interval

Lin

Wang

et al. 2021

Mathematics

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We face the problem to determine whether an algebraic polynomial is nonnegative in an interval the Yau Number Theoretic Conjecture and Yau Geometric Conjecture is proved. In this paper, we propose a new theorem to determine if an algebraic polynomial is nonnegative in an interval. It improves Wang-Yau Lemma for wider applications in light of Sturm’s Theorem. Many polynomials can use the new theorem but cannot use Sturm’s Theorem and Wang-Yau Lemma to judge whether they are nonnegative in an interval. New Theorem also performs better than Sturm’s Theorem when the number of terms and degree of polynomials increase. Main Theorem can be used for polynomials whose coefficients are parameters and to any interval we use. It helps us to find the roots of complicated polynomials. The problem of constructing nonnegative trigonometric polynomials in an interval is a classical, important problem and crucial to many research areas. We can convert a given trigonometric polynomial to an algebraic polynomial. Hence, our proposed new theorem affords a new way to solve this classical, important problem.

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“…al. [9] describes how determining the values of P n and Q n leads to insights in singularity theory.…”

Section: Introductionmentioning

confidence: 99%

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

Lin

Raghuvanshi

Yau

et al. 2018

Asian Journal of Mathematics

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The subject of counting positive lattice points in n-dimensional simplexes has interested mathematicians for decades due to its applications in singularity theory and number theory. Enumerating the lattice points in a right-angled simplex is equivalent to determining the geometric genus of an isolated singularity of a weighted homogeneous complex polynomial. It is also a method to shed insight into large gaps in the sequence of prime numbers. Seeking to contribute to these applications, in this paper, we prove the Yau Geometric Conjecture in six dimensions, a sharp upper bound for the number of positive lattice points in a six-dimensional tetrahedron. The main method of proof is summing existing sharp upper bounds for the number of points in 5-dimensional simplexes over the cross sections of the six-dimensional simplex. Our new results pave the way for the proof of a fully general sharp upper bound for the number of lattice points in a simplex. It also sheds new light on proving the Yau Geometric and Yau Number-Theoretic Conjectures in full generality.

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On a number theoretic conjecture on positive integral points in a 5-dimensional tetrahedron and a sharp estimate of the Dickman–De Bruijn function

Cited by 7 publications

References 25 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

Proposing a New Theorem to Determine If an Algebraic Polynomial Is Nonnegative in an Interval

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

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