×

The Frobenius problem, rational polytopes, and Fourier-Dedekind sums. (English) Zbl 1038.11026

The authors provide interesting links between rational polytopes, a linear Diophantine problem of Frobenius, and Dedekind-like finite Fourier series. They obtain a reciprocity law for these sums and use it to rederive Zagier’s reciprocity law for higher-dimensional Dedekind sums.

MSC:

11F20 Dedekind eta function, Dedekind sums
11D04 Linear Diophantine equations
11P21 Lattice points in specified regions
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Barvinok, A. I., Computing the Ehrhart polynomial of a convex lattice polytope, Discrete Comput. Geom., 12, 35-48 (1994) · Zbl 0804.52009
[2] Beck, M., Counting lattice points by means of the residue theorem, Ramanujan J., 4, 299-310 (2000) · Zbl 0988.11045
[3] Beck, M.; Gessel, I. M.; Komatsu, T., The polynomial part of a restricted partition function related to the Frobenius problem, Electron. J. Combin., 8 (2001) · Zbl 0982.05010
[4] Brauer, A.; Shockley, J. E., On a problem of Frobenius, J. Reine Angew. Math., 211, 215-220 (1962) · Zbl 0108.04604
[5] Brion, M., Points entiers dans les polyèdres convexes, Ann. Sci. École Norm. Sup (4), 21, 653-663 (1988) · Zbl 0667.52011
[6] Brion, M.; Vergne, M., Residue formulae, vector partition functions and lattice points in rational polytopes, J. Amer. Math. Soc., 10, 797-833 (1997) · Zbl 0926.52016
[7] Cappell, S. E.; Shaneson, J. L., Euler-Maclaurin expansions for lattices above dimension one, C. R. Acad. Sci. Paris Ser. I Math., 321, 885-890 (1995) · Zbl 0838.52018
[8] Davison, J. L., On the linear diophantine problem of Frobenius, J. Number Thoery, 48, 353-363 (1994) · Zbl 0805.11025
[9] Diaz, R.; Robins, S., The Erhart polynomial of a lattice polytope, Ann. Math., 145, 503-518 (1997) · Zbl 0874.52009
[10] Dieter, U., Das Verhalten der Kleinschen Funktionen \(logσ_{ g,h }(w_1, w_2)\) gegenüber Modultransformationen und verallgemeinerte Dedekindsche Summen, J. Reine Angew. Math., 201, 37-70 (1959) · Zbl 0085.02604
[11] Ehrhart, E., Sur un problème de géométrie diophantienne linéaire II, J. Reine Angew. Math., 227, 25-49 (1967)
[12] Erdös, P.; Graham, R. L., On a linear diophantine problem of Frobenius, Acta Arithm., 21, 399-408 (1972) · Zbl 0246.10010
[13] Gessel, I., Generating functions and generalized Dedekind sums, Electronic J. Combin., 4, R 11 (1997)
[14] Guillemin, V., Riemann-Roch for toric orbifolds, J. Differential Geom., 45, 53-73 (1997) · Zbl 0932.37039
[15] Johnson, S. M., A linear diophantine problem, Canad. J. Math., 12, 390-398 (1960) · Zbl 0096.02803
[16] Kannan, R., Lattice translates of a polytope and the Frobenius problem, Combinatorica, 12, 161-177 (1992) · Zbl 0753.11013
[17] Kantor, J.-M.; Khovanskii, A. G., Une application du Théorème de Riemann-Roch combinatoire au polynôme d’Ehrhart des polytopes entier de \(R^n\), C. R. Acad. Sci. Paris, Series I, 317, 501-507 (1993) · Zbl 0791.52012
[18] Khovanskii, A. G.; Pukhlikov, A. V., The Riemann-Roch theorem for integrals and sums of quasipolynomials on virtual polytopes, St. Petersburg Math. J., 4, 789-812 (1993) · Zbl 0798.52010
[19] Knuth, D. E., Notes on generalized Dedekind sums, Acta Arithm., 33, 297-325 (1977) · Zbl 0326.10007
[20] Macdonald, I. G., Polynomials associated with finite cell complexes, J. London Math. Soc., 4, 181-192 (1971) · Zbl 0216.45205
[21] Meyer, C., Über einige Anwendungen Dedekindscher Summen, J. Reine Angew. Math., 198, 143-203 (1957) · Zbl 0079.10303
[22] Rademacher, H., Some remarks on certain generalized Dedekind sums, Acta Arithm., 9, 97-105 (1964) · Zbl 0128.27101
[23] Rademacher, H.; Grosswald, E., Dedekind sums, Carus Mathematical Monographs (1972), The Mathematical Association of America: The Mathematical Association of America Washington · Zbl 0251.10020
[24] Robins, S., Generalized Dedekind η-products, Contemp. Math., 166, 119-128 (1994) · Zbl 0808.11031
[25] Rodseth, O. J., On a linear problem of Frobenius, J. Reine Angew. Math., 301, 171-178 (1978) · Zbl 0374.10011
[26] Rodseth, O. J., On a linear problem of Frobenius II, J. Reine Angew. Math., 307/308, 431-440 (1979) · Zbl 0395.10021
[27] Selmer, E. S., On the linear diophantine problem of Frobenius, J. Reine Angew. Math., 293/294, 1-17 (1977) · Zbl 0349.10009
[28] Sylvester, J. J., Mathematical questions with their solutions, Ed. Times, 41, 171-178 (1884)
[29] Vitek, Y., Bounds for a linear diophantine problem of Frobenius, J. London Math. Soc. (2), 10, 390-398 (1975)
[30] Zagier, D., Higher dimensional Dedekind sums, Math. Ann., 202, 149-172 (1973) · Zbl 0237.10025
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.