New Subexponential Fewnomial Hypersurface Bounds
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Suppose c_1,...,c_n+k are real numbers, {a_1,...,a_n+k}⊂R^n is a set of points not all lying in the same affine hyperplane, y∈R^n, a_j· y denotes the standard real inner product of a_j and y, and we set g(y):=∑^n+k_j=1 c_j e^a_j· y. We prove that, for generic c_j, the number of connected components of the real zero set of g is O(n^2+√(2)^k^2(n+2)^k-2). The best previous upper bounds, when restricted to the special case k=3 and counting just the non-compact components, were already exponential in n.
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