×

On the polynomial sharp upper estimate conjecture in 7-dimensional simplex. (English) Zbl 1396.11119

Summary: Because of its importance in number theory and singularity theory, the problem of finding a polynomial sharp upper estimate of the number of positive integral points in an \(n\)-dimensional (\(n \geq 3\)) polyhedron has received attention by a lot of mathematicians. The first named author proposed the Number Theoretic Conjecture for the upper estimate. The previous results on the Number Theoretic Conjecture in low dimension cases (\(n < 7\)) are proved by using the sharp GLY conjecture which is true only for low dimensional case. Thus the proof cannot be generalized to high dimension. In this paper, we offer a uniform approach to prove the Number Theoretic Conjecture for all dimensions by simply using the induction method and the Yau-Zhang [S. T. Yau and L. Zhang, Math. Res. Lett. 13, No. 5-6, 911–921 (2006; Zbl 1185.11062)] estimates (see Lemmas 2.3-2.5). As a result, the Number Theoretic Conjecture is proven for \(n = 7\). An important estimate for all dimensions is also obtained (Prepositions 3.1 and 3.2) which will be useful to prove the general case of the Number Theoretic Conjecture. As an application, we give a sharper estimate of the Dickman-De Bruijn function \(\Psi(x, y)\) for \(5 \leq y < 19\), compared with the result obtained by V. Ennola [Ann. Acad. Sci. Fenn., Ser. A I 440, 16 p. (1969; Zbl 0174.33903)].

MSC:

11P21 Lattice points in specified regions
11Y99 Computational number theory
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Chen, I.; Lin, K.-P.; Yau, S. S.-T.; Zuo, H. Q., Coordinate-free characterization of homogeneous polynomials with isolated singularities, Comm. Anal. Geom., 19, 4, 661-704 (2011) · Zbl 1246.32030
[2] Durfee, A. H., The signature of smoothings of complex surface singularities, Math. Ann., 232, 1, 85-98 (1978) · Zbl 0346.32016
[3] Ehrhart, E., Sur un problème de gèomètrie diophantienne linèaire II, J. Reine Angew. Math., 227, 25-49 (1967)
[4] Ennola, V., On Numbers with Small Prime Divisors, Ann. Acad. Sci. Fenn. Ser. AI, vol. 440 (1969), 16 pp · Zbl 0174.33903
[5] Liang, A.; Yau, S. S.-T.; Zuo, H. Q., A sharp estimate of positive integral points in 6-dimensional polyhedra and a sharp estimate of smooth numbers, Sci. China Math. (2015), in press
[6] Lin, K.-P.; Yau, S. S.-T., Counting the number of integral points in general n-dimensional tetrahedra and Bernoulli polynomials, Canad. Math. Bull., 46, 2, 229-241 (2003) · Zbl 1056.11054
[7] Lin, K.-P.; Luo, X.; Yau, S. S.-T.; Zuo, H. Q., On number theoretic conjecture of positive integral points in 5-dimensional tetrahedron and a sharp estimation of Dickman-De Bruijn function, J. Eur. Math. Soc., 16, 90-102 (2014)
[8] Luo, X.; Yau, S. S.-T.; Zuo, H. Q., A sharp estimate of Dickman-De Bruijn function and a sharp polynomial estimate of positive integral points in 4-dimension tetrahedron, Math. Nachr., 288, 1, 61-75 (2015) · Zbl 1386.11102
[9] Merle, M.; Teissier, B., Conditions d’adjonction, d’après du val, (Séminaire sur les Singularités des Surfaces (1980), Springer), 229-245 · Zbl 0461.14009
[10] Milnor, J.; Orlik, P., Isolated singularities defined by weighted homogeneous polynomials, Topology, 9, 4, 385-393 (1970) · Zbl 0204.56503
[11] Mordell, L. J., Lattice points in a tetrahedron and generalize Dedekind sums, J. Indian Math., 15, 41-46 (1951) · Zbl 0043.05101
[12] Pomerance, C., The role of smooth numbers in number theoretic algorithms, (ICM. ICM, Zürich (1994)), 411-422 · Zbl 0854.11047
[13] Pommersheim, J. E., Toric varieties, lattice points and Dedekind sums, Math. Ann., 295, 1, 1-24 (1993) · Zbl 0789.14043
[14] Saito, K., Quasihomogene isolierte singularitäten von hyperflächen, Invent. Math., 14, 2, 123-142 (1971) · Zbl 0224.32011
[15] van der Waerden, B. L., Modern Algebra, vol. I (1950), Frederick Unger Publishing Co. · Zbl 0037.01903
[16] Wang, X.-J.; Yau, S. S.-T., On the GLY conjecture of upper estimate of positive integral points in real right-angled simplices, J. Number Theory, 122, 1, 184-210 (2007) · Zbl 1115.11062
[17] Xu, Y.-J.; Yau, S. S.-T., A sharp estimate of the number of integral points in a tetrahedron, J. Reine Angew. Math., 423, 199-219 (1992) · Zbl 0734.11048
[18] Xu, Y.-J.; Yau, S. S.-T., A sharp estimate of the number of integral points in a 4-dimensional tetrahedra, J. Reine Angew. Math., 473, 69-86 (1996)
[19] Yau, S. S.-T.; Zhang, L.-T., An upper estimate of integral points in real simplices with an application to singularity theory, Math. Res. Lett., 13, 911-921 (2006) · Zbl 1185.11062
[20] Yau, S. S.-T.; Zuo, H. Q., Complete characterization of isolated homogeneous hypersurface singularities, Pacific J. Math., 260, 1, 245-255 (2012) · Zbl 1276.32022
[21] Yau, S. S.-T.; Zuo, H. Q., Lower estimate of Milnor number and characterization of isolated homogeneous hypersurface singularities, Pacific J. Math., 273, 1, 213-224 (2015)
[22] Yau, S. S.-T.; Zhao, L.; Zuo, H. Q., Biggest sharp polynomial estimate of integral points in right-angled simplices, (Topology of Algebraic Varieties and Singularities. Topology of Algebraic Varieties and Singularities, Contemp. Math., vol. 538 (2011), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 433-467 · Zbl 1284.11129
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.