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Laurent determinants and arrangements of hyperplane amoebas. (English) Zbl 1002.32018
Author’s introduction: The notion of amoebas was introduced by I. M. Gelfand, M. M. Kapranov, and A. V. Zelevinsky in ‘Discriminants, resultants and multidimensional determinants’ (1994; Zbl 0827.14036). Given a Laurent polynomial $$f$$ its amoeba $${\mathcal A}_f$$ is the image of the hypersurface $${\mathcal Z}_f=f^{-1}(0)$$ under the map $$(z_1,\dots,z_n) \mapsto( \log |z_1|, \dots,\log |z_n|)$$. It will typically be a semianalytic closed subset of $$\mathbb{R}^n$$ with tentacle-like asymptotes going off to infinity and separating the connected components of the complement $$^c{\mathcal A}_f$$. These components are convex and they reflect the structure of the Newton polytope $${\mathcal N}_f$$ of the Laurent polynomial $$f$$. Furthermore, each such component corresponds to a specific Laurent series development of the rational function $$1/f$$. The problem of finding and describing the connected components of $$^c {\mathcal A}_f$$ was posed in the paper cited above.
In this paper we introduce what we call the order of a complement component, and we show that it provides a bijection between the family of components and a subset of $${\mathcal N}_f\cap \mathbb{Z}^n$$. This implies in particular that the number of connected components of $$^c{\mathcal A}_f$$ is at most equal to the number of integer points in the Newton polytope. We then go on to introduce a certain matrix of Laurent coefficients of $$1/f$$. Even though the individual Laurent coefficients may be unwidely hypergeometric functions, the (square of the) determinant of this matrix, which we call the Laurent determinant of $$f$$, appears to have a tractable structure.
We devote the last part of the paper to the special situation where $$f$$ is a polynomial that factors into linear forms. Its zero set is then a union of hyperplanes, and consequently the amoeba is a union, or an arrangement, of hyperplane amoebas. It is proved that when the coefficients of the linear functions lie outside a certain secondary amoeba, the number of components of the complement $$^c{\mathcal A}_f$$ is maximal, that is, equal to the number of integer points in the Newton polytope $${\mathcal N}_f$$. We are also able in this case to compute the Laurent determinant of $$f$$ explicitly, and it turns out to be exactly equal to the reciprocal of the polynomial defining the aforesaid secondary amoeba.

##### MSC:
 32S22 Relations with arrangements of hyperplanes
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##### References:
 [1] Arnold, V.; Gusein-Zade, S.; Varchenko, A., Singularities of differentiable maps, Monographs in mathematics, 83, (1988), Birkhäuser Boston [2] Batyrev, V., Variations of the mixed Hodge structure of affine hypersurfaces in algebraic tori, Duke math. J., 69, 349-409, (1993) · Zbl 0812.14035 [3] Curtis, C.; Reiner, I., Representation theory of finite groups and associative algebras, Pure and applied mathematics, (1962), Wiley New York · Zbl 0131.25601 [4] Danzer, L.; Grünbaum, B.; Klee, V., Helly’s theorem and its relatives, Convexity, Proceedings of symposia in pure mathematics, 7, (1963), American Mathematical Society Providence, p. 101-180 · Zbl 0132.17401 [5] Forsberg, M., Amoebas and Laurent series, (1998), Royal Institute of Technology Stockholm [6] Fotiadi, D.; Froissart, M.; Lascoux, J.; Pham, F., Applications of an isotopy theorem, Topology, 4, 159-191, (1965) · Zbl 0173.09301 [7] Gelfand, I.; Kapranov, M.; Zelevinsky, A., Discriminants, resultants and multidimensional determinants, (1994), Birkhäuser Boston · Zbl 0827.14036 [8] Mkrtchian, M.; Yuzhakov, A., The Newton polytope and the Laurent series of rational functions of n variables, Izv. akad. nauk armssr, 17, 99-105, (1982) [9] Rudin, W., Function theory in polydiscs, Mathematics lecture note series, (1969), Benjamin New York/Amsterdam · Zbl 0177.34101 [10] Tsikh, A., Multidimensional residues and their applications, Translations of mathematical monographs, 103, (1992), American Mathematical Society Providence [11] Ziegler, G., Lectures on polytopes, Graduate texts in mathematics, (1995), Springer New York/Berlin · Zbl 0823.52002
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