Optimal Bounds for the Number of Pieces of Near-Circuit Hypersurfaces
Weixun Deng, J. Maurice Rojas, Cordelia Russell
TL;DR
The paper resolves an open Fewnomial Theory question by proving that honest $n$-variate $(n+3)$-nomials with a positive-volume Newton polytope have at most $3$ connected components in their positive real zero set $Z_+(f)$. The authors develop and deploy a sophisticated framework based on real exponential sums, signed $\mathcal{A}$-discriminants, and their reduced contours, together with Gale duals and Horn–Kapranov uniformization, to reduce the problem to a planar discriminant curve $\Gamma_\varepsilon(\mathcal{A},B)$ whose cusps govern isotopy types. Central to the argument are Morse-theoretic analyses of Hessians along discriminant curves, enabling precise tracking of how the number of pieces changes (or remains invariant) as coefficients vary across chambers; the changes are shown to occur only at cusps and are tightly controlled by Hessian indices. The results extend to exponential sums with real exponents and provide a toolkit—cuspidal structure, reduced chamber stratification, and Hessian-sign analysis—that can be applied to broader quantitative geometric problems in real algebraic geometry.
Abstract
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 $k\!=\!3$, 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.