no. 3
Binomial residues
[Résidus binomiaux]
Cattani, Eduardo1 ; Dickenstein, Alicia2 ; Sturmfels, Bernd3
1 University of Massachusetts, Department of Mathematics and Statistics, Amherst MA 01003 (USA)
2 Universidad de Buenos Aires, Departamento de Matematica, FCEyN (1428), Buenos Aires (Argentine)
3 University of California, Department of Mathematics, Berkeley CA 94720(USA)
Annales de l'Institut Fourier, Tome 52 (2002) no. 3, pp. 687-708.

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.

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.

DOI : 10.5802/aif.1898
Classification : 05B35, 14M25, 32A27
Keywords: binomial residues, hypergeometric functions, Lawrence configurations
Mot clés : résidus binomiaux, fonctions hypergéométriques, configurations de Lawrence

Cattani, Eduardo 1 ; Dickenstein, Alicia 2 ; Sturmfels, Bernd 3

1 University of Massachusetts, Department of Mathematics and Statistics, Amherst MA 01003 (USA)
2 Universidad de Buenos Aires, Departamento de Matematica, FCEyN (1428), Buenos Aires (Argentine)
3 University of California, Department of Mathematics, Berkeley CA 94720(USA) 
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Cattani, Eduardo; Dickenstein, Alicia; Sturmfels, Bernd. Binomial residues. Annales de l'Institut Fourier, Tome 52 (2002) no. 3, pp. 687-708. doi : 10.5802/aif.1898. https://aif.centre-mersenne.org/articles/10.5802/aif.1898/

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

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

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

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

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

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

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

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

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

[10] D. Cox Toric residues, Arkiv för Matematik, Volume 34 (1996), pp. 73-96 | DOI | MR | Zbl

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

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

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

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

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

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

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

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

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

[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), Volume Vol. I, II (1991), pp. 281-300 | Zbl

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

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