Binomial residues
Annales de l'Institut Fourier, Volume 52 (2002) no. 3, p. 687-708

A binomial residue is a rational function defined by a hypergeometric integral whose kernel is singular along binomial divisors. Binomial residues provide an integral representation for rational solutions of A-hypergeometric systems of Lawrence type. The space of binomial residues of a given degree, modulo those which are polynomial in some variable, has dimension equal to the Euler characteristic of the matroid associated with A.

Un résidu binomial est une fonction rationnelle définie par une intégrale hypergéométrique ayant un noyau singulier le long d’un diviseur binomial. Les résidus binomiaux donnent une représentation intégrale des solutions rationnelles des systèmes A-hypergéométriques du type de Lawrence. L’espace des résidus binomiaux d’un degré donné, modulo ceux qui dépendent polynomialement d’une des variables, a sa dimension égale à la caractéristique d’Euler du matroïde associé à A.

DOI : https://doi.org/10.5802/aif.1898
Classification:  05B35,  14M25,  32A27
Keywords: binomial residues, hypergeometric functions, Lawrence configurations
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     author = {Cattani, Eduardo and Dickenstein, Alicia and Sturmfels, Bernd},
     title = {Binomial residues},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {52},
     number = {3},
     year = {2002},
     pages = {687-708},
     doi = {10.5802/aif.1898},
     zbl = {1015.32007},
     mrnumber = {1907384},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2002__52_3_687_0}
}
Binomial residues. Annales de l'Institut Fourier, Volume 52 (2002) no. 3, pp. 687-708. doi : 10.5802/aif.1898. http://www.numdam.org/item/AIF_2002__52_3_687_0/

[1] V. Batyrev; D. Cox On the Hodge structure of projective hypersurfaces in toric varieties, Duke Math., Tome 75 (1994), pp. 293-338 | MR 1290195 | Zbl 0851.14021

[2] A. Björner The homology and shellability of matroids and geometric lattices, Cambridge University Press, Matroid Applications (1992) | MR 1165544 | Zbl 0772.05027

[3] A. Björner; M.Las Vergnas; B.Sturmfels; N. White; G. Ziegler Oriented Matroids, Cambridge University Press (1993) | MR 1226888 | Zbl 0773.52001

[4] M. Brion; M. Vergne Arrangement of hyperplanes. I. Rational functions and Jeffrey-Kirwan residue, Ann. Sci. École Norm. Sup., Tome 32 (1999), pp. 715-741 | Numdam | MR 1710758 | Zbl 0945.32003

[5] E. Cattani; D. Cox; A. Dickenstein Residues in toric varieties, Compositio Mathematica, Tome 108 (1997), pp. 35-76 | Article | MR 1458757 | Zbl 0883.14029

[6] E. Cattani; A. Dickenstein A global view of residues in the torus, Journal of Pure and Applied Algebra, Tome 117 \& 118 (1997), pp. 119-144 | Article | MR 1457836 | Zbl 0899.14024

[7] E. Cattani; A. Dickenstein; B. Sturmfels Residues and resultants, J. Math. Sci. Univ. Tokyo, Tome 5 (1998), pp. 119-148 | MR 1617074 | Zbl 0933.14033

[8] E. Cattani; A. Dickenstein; B. Sturmfels Rational hypergeometric functions, Compositio Mathematica, Tome 128 (2001), pp. 217-240 | Article | MR 1850183 | Zbl 0990.33013

[9] D. Cox The homogeneous coordinate ring of a toric variety, Journal of Algebraic Geometry, Tome 4 (1995), pp. 17-50 | MR 1299003 | Zbl 0846.14032

[10] D. Cox Toric residues, Arkiv för Matematik, Tome 34 (1996), pp. 73-96 | Article | MR 1396624 | Zbl 0904.14029

[11] I. M. Gel'Fand; A. Zelevinsky; M. Kapranov Hypergeometric functions and toral manifolds, Functional Analysis and its Appl., Tome 23 (1989), pp. 94-106 | Article | MR 1011353 | Zbl 0721.33006

[12] I. M. Gel'Fand; M. Kapranov; A. Zelevinsky Generalized Euler integrals and 𝒜-hypergeometric functions, Advances in Math., Tome 84 (1990), pp. 255-271 | Article | MR 1080980 | Zbl 0741.33011

[13] P. Griffiths; J. Harris Principles of Algebraic Geometry, John Wiley \& Sons, New York (1978) | MR 507725 | Zbl 0408.14001

[14] J. Kaneko The Gauss-Manin connection of the integral of the deformed difference product, Duke Math. J., Tome 92 (1998), pp. 355-379 | MR 1612801 | Zbl 0947.32009

[15] I. Novik; A. Postnikov; B.Sturmfels Syzygies of oriented matroids, Duke Math. J., Tome 111 (2002), pp. 287-317 | Article | MR 1882136 | Zbl 1022.13002

[16] P. Orlik; H. Terao Arrangements of Hyperplanes, Springer-Verlag, Heidelberg, Grundlehren der mathematisches Wissenchaften, Tome Volume 300 (1992) | MR 1217488 | Zbl 0757.55001

[17] B. Sturmfels Gröbner Bases and Convex Polytopes, American Mathematical Society, Providence (1995) | MR 1363949 | Zbl 0856.13020

[18] M. Saito; B. Sturmfels; And N. Takayama Gröbner Deformations of Hypergeometric Differential Equations, Springer-Verlag, Heidelberg, Algorithms and Computation in Mathematics, Tome Volume 6 (2000) | MR 1734566 | Zbl 0946.13021

[19] A. Tsikh Multidimensional Residues and Their Applications, American Math. Society, Providence (1992) | MR 1181199 | Zbl 0758.32001

[20] A. Varchenko Multidimensional hypergeometric functions in conformal field theory, algebraic K-theory, algebraic geometry, Proceedings of the International Congress of Mathematicians, (Kyoto, 1990) (Math. Soc. Japan Tokyo) Tome Vol. I, II (1991), pp. 281-300 | Zbl 0747.33002

[21] T. Zaslavsky Facing up to arrangements: face-count formulas for partitions of space by hyperplanes, Memoirs of the AMS, Tome 1 (1975) no. 154 | MR 357135 | Zbl 0296.50010