2019/04/03 by Yifan Jing, Jing, Yifan, Souktik Roy +3
Computer Science · Mathematics · #03C64 #05D10 #Algebraic Geometry and Number Theory #Coding theory and cryptography #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory #Logic (math.LO) #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1904.01715
openalex publication_date 2019/04/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We begin a generalized study of sum-product type phenomenon in different fields by considering pairs P(x,y) and Q(x,y) of two variable polynomials that simultaneously exhibit small symmetric expansion. Our first result is that such P(x,y) and Q(x,y) over ℝ and ℂ have very similar structure, obtained by employing semi-algebraic geometry/o-minimality. Then using model-theoretic transfer and basic Galois theory we deduce results for fields of characteristic 0 and characteristic p when p is large. We obtain as corollaries a generalization of Elekes-Rónyai type structural results to arbitrary characteristic 0 fields, and a strengthening of these classic results in a symmetric case of natural interest. We note a related bound of 5/4 in the exponent for the sum-product problem in finite fields of large characteristic, although a lower bound for this characteristic cannot be computed from our methods.