A-Discriminants for Complex Exponents, and Counting Real Isotopy Types
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We extend the definition of A-discriminant varieties, and Kapranov's parametrization of A-discriminant varieties, to complex exponents. As an application, we study the special case where A is a fixed real n× (n+3) matrix whose columns form the spectrum of an n-variate exponential sum g with fixed sign vector for its coefficients: We prove that the number of possible isotopy types for the real zero set of g is O(n^2). The best previous upper bound was 2^O(n^4). Along the way, we also show that the singular loci of our generalized A-discriminants are images of low-degree algebraic sets under certain analytic maps.
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