New Subexponential Fewnomial Hypersurface Bounds
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abstract
Suppose $c_1,ldots,c_{n+k}$ are real numbers, ${a_1,ldots,a_{n+k}}!subset!mathbb{R}^n$ is a set of points not all lying in the same affine hyperplane, $y!in!mathbb{R}^n$, $a_jcdot y$ denotes the standard real inner product of $a_j$ and $y$, and we set $g(y)!:=!sum^{n+k}_{j=1} c_j e^{a_jcdot y}$. We prove that, for generic $c_j$, the number of connected components of the real zero set of $g$ is $O!left(n^2+sqrt{2}^{k^2}(n+2)^{k-2} ight)$. The best previous upper bounds, when restricted to the special case $k!=!3$ and counting just the non-compact components, were already exponential in $n$.
