Laurent determinants and arrangements of hyperplane amoebas.

*(English)*Zbl 1002.32018Author’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.

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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\textit{M. Forsberg} et al., Adv. Math. 151, No. 1, 45--70 (2000; Zbl 1002.32018)

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