Optimal Bounds for the Number of Pieces of Near-Circuit Hypersurfaces

Suppose $f$ is a polynomial in $n$ variables with real coefficients, exactly $n+k$ monomial terms, and Newton polytope of positive volume. Estimating the number of connected components of the positive zero set of $f$ is a fundamental problem in real algebraic geometry, with applications in computational complexity and topology. We prove that the number of connected components is at most $3$ when $k\text{\hspace{0.17em}⁣}=\text{\hspace{0.17em}⁣}3$, settling an open question from Fewnomial Theory. Our results also extend to exponential sums with real exponents. A key contribution here is a deeper analysis of the underlying $\mathcal{A}$-discriminant curves, which should be of use for other quantitative geometric problems.