The authors study singularities of hypergeometric functions which are defined by means of analytic continuation of hypergeometric series. A hypergeometric series \(y(x)\) satisfies the so-called Horn hypergeometric system \[ x_iP_i(\theta)y(x)=Q_i(\theta)y(x),\quad i =1,\dots,n. \tag{1} \] Here \(P_i\) and \(Q_i\) are nonzero polynomials depending on the vector differential operator \(\theta = (\theta_1,\dots,\theta_n)\), \(\theta_i = x_i\partial/\partial x_i\). The nonconfluency of a hypergeometric series or the system (1) means that the polynomials \(P_i\) and \(Q_i\) are of the same degree: \(\text{deg}\,P_i=\text{deg}\,Q_i\), \(i =1,\dots,n\). These conditions can be expressed in terms of the Ore-Sato coefficient of a hypergeometric series satisfying the system (1). Historically the Gauss hypergeometric differential equation was the first one to be studied in detail due to the remarkable fact that any linear homogeneous differential equation of order two with three regular singularities can be reduced to it. The singularities of the Gauss equation are 0, 1 and \(\infty\). The generalized ordinary hypergeometric differential equation which is a special case of the nonconfluent system (1) corresponding to \(n=1\) also has three singular points, namely 0, \(t\) and \(\infty\), where \(t\) is the quotient of the coefficients in the leading terms in the polynomials \(P_1\) and \(Q_1\). Thus the singular set of an ordinary hypergeometric differential equation is minimal in the following precise sense: There exist only two circular domains, namely \(\{0< | x| <| t| \}\) and \(\{| t| < | x| < \infty\}\), in which any solution to the equation can be represented as a Laurent series with the center at the origin (in the nonresonant case) or as a linear combination of the products of Laurent series and powers of \(\log x\) (in the resonant case). It turns out that algebraic singularities of the system of partial differential equations (1) enjoy a multi-dimensional analog of this minimality property. This property is most conveniently formulated in the language of so-called amoebas, a terminology introduced by Gelfand, Kapranov and Zelevinskij. The amoeba of an algebraic set \(\mathcal R = \{R(x) = 0\}\) is defined to be its image under the mapping \(\text{Log} : (x_i,\dots, x_n)\mapsto (\log | x_1| ,\dots, \log | x_n| )\).
The complement of an amoeba consists of a finite number of convex connected components which correspond to domains of convergence of the Laurent series expansions of rational functions with denominator \(\mathcal R\). The number of such components cannot be smaller than the number of vertices of the Newton polytope of the polynomial \(R(x)\). If these two numbers are equal, then we say that the amoeba is solid. The authors prove the following theorem: The singular hypersurface of any nonconfluent hypergeometric function has a solid amoeba. A hypergeometric function satisfying the Gelfand-Kapranov-Zelevinskij system of equations has singularities along the zero locus of the corresponding principal \(\mathcal A\)-determinant. Using the above theorem the authors arrive at the following corollary:
The zero set of any principal \(\mathcal A\)-determinant has a solid amoeba. This corollary implies in particular that the amoeba of the classical discriminant of a general algebraic equation is solid. Let us also mention the following results in this paper: Any meromorphic nonconfluent hypergeometric function is rational. The authors also study the problem of describing the class of rational hypergeometric functions. It is given a necessary condition for the Horn system to possess a rational solution. Only very few rational functions are hypergeometric. The class of rational hypergeometric functions that is described in this proposition consists of those which are contiguous to Bergman kernels of complex ellipsoidal domains. The proofs of the main results in the paper use the notions of the support and the fan of a hypergeometric series, some facts from toric geometry and the two-sided Abel lemma, which is proved in this paper.