On the Sharpness of fewnomial bound and the number of components of a fewnomial hypersurface

Betti number bounds for fewnomial hypersurfaces via stratified Morse theory

We use stratified Morse theory for a manifold with corners to give a new bound for the sum of the Betti numbers of a hypersurface in R^n_> defined by a polynomial with n+l+1 terms.

Dense Fewnomials

We derive new bounds of fewnomial type for the number of real solutions to systems of polynomials that have structure intermediate between fewnomials and generic (dense) polynomials. This uses a modified version of Gale duality for polynomial systems. We also use stratified Morse theory to bound the total Betti number of a hypersurface defined by such a dense fewnomial. These bounds contain and generalize previous bounds for ordinary fewnomials obtained by Bates, Bertrand, Bihan, and Sottile.

New Subexponential Fewnomial Hypersurface Bounds

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 …

少项超曲面连通分量数的新上界

New bounds for the number of connected components of fewnomial hypersurfaces

本文针对正象限中由少项实多项式定义的光滑超曲面，给出了连通分量数的新上界，并在二维情形构造达到上界的显式例子，证明结果精确。

从Gale对偶多项式系统获得新的稀少多项式上界

New fewnomial upper bounds from Gale dual polynomial systems

本文通过Gale对偶多项式系统的方法，推导出稀少多项式系统的解数上界，并研究了超曲面的连通分支数。

Counting Isolated Roots of Trinomial Systems in the Plane and Beyond

We prove that any pair of bivariate trinomials has at most 5 isolated roots in the positive quadrant. The best previous upper bounds independent of the polynomial degrees counted only non-degenerate roots and even then gave much larger bounds, e.g., 248832 via a famous general result of Khovanski. Our bound is sharp, allows real exponents, and extends to certain systems of n-variate fewnomials, giving improvements over earlier bounds by a factor exponential in the number of monomials. We also derive new bounds on the number of real connected components of fewnomial hypersurfaces.