New bounds for the number of connected components of fewnomial hypersurfaces

Abstract

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 $8(d+1{)}^{k-1}{2}^{\left(\genfrac{}{}{0ex}{}{k-1}{2}\right)}$, 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.

References in corpus (2)

• Patchworking real algebraic varieties
• Some geometrical aspects of control points for toric patches