2025/02/14 by Weixun Deng, Deng, Weixun, J. Maurice Rojas +3
Computer Science · Mathematics · #Algebraic Geometry (math.AG) #Computational Geometry and Mesh Generation #FOS: Mathematics #Point processes and geometric inequalities #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2502.10590
openalex publication_date 2025/02/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 A-discriminant curves, which should be of use for other quantitative geometric problems.