New Subexponential Fewnomial Hypersurface Bounds

Suppose $c_1,\dots ,c_{n+k}$ are real numbers, $\{a_1,\dots ,a_{n+k}\}\text{\hspace{0.17em}⁣}\subset \text{\hspace{0.17em}⁣}{\mathbb{R}}^n$ is a set of points not all lying in the same affine hyperplane, $y\text{\hspace{0.17em}⁣}\in \text{\hspace{0.17em}⁣}{\mathbb{R}}^n$, $a_j\cdot y$ denotes the standard real inner product of $a_j$ and $y$, and we set $g(y)\text{\hspace{0.17em}⁣}:=\text{\hspace{0.17em}⁣}{\sum }_{j=1}^{n+k}{c}_j{e}^{a_j\cdot y}$. We prove that, for generic $c_j$, the number of connected components of the real zero set of $g$ is $O\text{\hspace{0.17em}⁣}(n^2+{\sqrt{2}}^{k^2}(n+2{)}^{k-2})$. The best previous upper bounds, when restricted to the special case $k\text{\hspace{0.17em}⁣}=\text{\hspace{0.17em}⁣}3$ and counting just the non-compact components, were already exponential in $n$.