New bounds for the number of connected components of fewnomial hypersurfaces

We prove that the number of connected components of a smooth hypersurface in the positive orthant of $\mathbb{R}^n$ defined by a real polynomial with $d + k + 1$ monomials, where $d$ is the dimension of the affine span of the exponent vectors, is smaller than or equal to $8(d+1)^{k-1} 2^{k-1 \choose 2}$, improving the previously known bounds. We refine this bound for $k = 2$ by showing that a smooth hypersurface defined by a real polynomial with $d+3$ monomials in $n$ variables has at most $\lfloor(d-1)/2\rfloor + 3$ connected components in the positive orthant of $\mathbb{R}^n$. We present an explicit polynomial in $2$ variables with $5$ monomials which defines a curve with three connected components in the positive orthant, showing that our bound is sharp for $d = 2$ (and any $n$). Our results hold for polynomials with real exponent vectors.