New Subexponential Fewnomial Hypersurface Bounds

TL;DR

This paper establishes new subexponential bounds on the number of connected components of certain exponential sum hypersurfaces, improving upon previous exponential bounds for specific cases.

Contribution

It provides the first subexponential upper bounds for the topology of real zero sets of exponential sums with a fixed number of terms.

Findings

Number of connected components is bounded by O(n^2 + 2^{k^2/2}(n+2)^{k-2})

Previous bounds were exponential in n for the case k=3

Results apply to generic coefficient choices

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, $y\!\in\!\mathbb{R}^n$, $a_j\cdot y$ denotes the standard real inner product of $a_j$ and $y$, and we set $g(y)\!:=\!\sum^{n+k}_{j=1} c_j e^{a_j\cdot y}$. We prove that, for generic $c_j$, the number of connected components of the real zero set of $g$ is $O\!\left(n^2+\sqrt{2}^{k^2}(n+2)^{k-2}\right)$. The best previous upper bounds, when restricted to the special case $k\!=\!3$ and counting just the non-compact components, were already exponential in $n$.