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

TL;DR

This paper establishes a tight upper bound of three on the number of connected components of the positive zero set for certain sparse polynomials, advancing understanding in real algebraic geometry and exponential sums.

Contribution

It proves the first nontrivial bound for the number of components when the polynomial has exactly three additional monomials, extending Fewnomial Theory.

Findings

Number of components is at most 3 for polynomials with n+3 monomials.

Extends results to exponential sums with real exponents.

Provides a detailed analysis of $\\mathcal{A}$-discriminant curves.

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.