Suppose c_1,\ldots,c_{n+k} are real numbers, \{a_1,\ldots,a_{n+k}\}\!\subset\!R^n is a set of points not all lying in the same affine hyperplane, y\!\in\!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+2^{k^2}(n+2)^{k-2}\right). The best previous upper bound, just for the number of non-compact components, in the special case where k\!=\!3 and \{a_1, \ldots,a_{n+k\}\!\subset\!Z^n, was }  exponential in n.
Suppose c_1,\ldots,c_{n+k} are real numbers, \{a_1,\ldots,a_{n+k}\}\!\subset\!R^n is a set of points not all lying in the same affine hyperplane, y\!\in\!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+2^{k^2}(n+2)^{k-2}\right). The best previous upper bounds, when restricted to the special case k\!=\!3 and \}\!\subset\!Z^n, was } counting just the non-compact components, were already exponential in n.
R
bound, just for the number of non-compact components, in the special case where
bounds, when restricted to the special case
D
\{a_1, \ldots,a_{n+k
A
counting just the non-compact components, were already