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

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

doi:10.1002/mana.201010081

Cited by 5 publications

(5 citation statements)

References 19 publications

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“…By the previous works of Xu and Yau [26], [28], it was shown that the numbertheoretic conjecture is true for n = 3. The case n = 4 has been shown in our previous work [18]. The purpose of this paper is to prove that the number-theoretic conjecture is true for n = 5.…”

Section: Introductionmentioning

confidence: 67%

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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

Lin¹,

Luo²,

Yau³

et al. 2014

J. Eur. Math. Soc.

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It is well known that getting an estimate of the number of integral points in right-angled simplices is equivalent to getting an estimate of the Dickman-de Bruijn function ψ(x, y) which is the number of positive integers ≤ x and free of prime factors > y. Motivated by the Yau Geometric Conjecture, the third author formulated a number-theoretic conjecture which gives a sharp polynomial upper estimate on the number of positive integral points in n-dimensional (n ≥ 3) real right-angled simplices. In this paper, we prove this conjecture for n = 5. As an application, we give a sharp estimate of the Dickman-de Bruijn function ψ(x, y) for 5 ≤ y < 13.

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Section: Introductionmentioning

confidence: 67%

“…In detail, let p 1 < • • • < p 5 be 5 primes ≤ y. Clearly p l Cases (i) and (ii) have been proven in [18]. For (iii), we have five prime numbers p 1 = 2,…”

Section: Proof Of Theorem 13mentioning

confidence: 97%

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

Lin¹,

Luo²,

Yau³

et al. 2014

J. Eur. Math. Soc.

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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: 79%

“…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: 90%

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

2015

Self Cite

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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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“…[9,10]). For more details concerning a recent progress in this rapidly growing area of counting positive lattice points in n-dimensional simplexes, the interested reader is referred to [11][12][13]. Stephen S.-T. Yau proposes Yau Number Theoretic Conjecture and Yau Geometric Conjecture (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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A sharp polynomial estimate of positive integral points in a 4-dimensional tetrahedron and a sharp estimate of the Dickman-de Bruijn function

Cited by 5 publications

References 19 publications

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

On a number theoretic conjecture on positive integral points in a 5-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

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

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