New bounds for the number of connected components of fewnomial hypersurfaces

TL;DR

This paper establishes new upper bounds on the number of connected components of certain hypersurfaces defined by fewnomial polynomials in the positive orthant, refining previous bounds and demonstrating sharpness in specific cases.

Contribution

It introduces improved bounds for the number of connected components of fewnomial hypersurfaces, especially for the case when the number of monomials is small, and provides explicit examples showing the bounds are sharp.

Findings

New upper bounds for connected components of hypersurfaces with few monomials.

Refined bounds specifically for polynomials with $d+3$ monomials.

Explicit example of a polynomial with 5 monomials having 3 connected components.

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 $8(d+1)^{k-1} 2^{k-1 \choose 2}$, 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.