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Title: Semialgebraic Methods and Generalized Sum-Product Phenomena
Authors: Jing, Y
Roy, S
Tran, CM 
Issue Date: 1-Jan-2022
Citation: Jing, Y, Roy, S, Tran, CM (2022-01-01). Semialgebraic Methods and Generalized Sum-Product Phenomena. Discrete Analysis 2022. ScholarBank@NUS Repository.
Abstract: For a bivariate [Formula Presented] our first result shows that for all finite A ⊆ R, |P(A,A)| ≥ α|A|5/4 with α = α(degP) ∈ R>0 unless P(x,y) = f (γu(x)+δu(y)) or P(x,y) = f (um(x)un(y)) for some univariate f,u ∈ R[t] \R, constants γ,δ ∈R≠0, and m,n ∈N≥1. This resolves the symmetric nonexpanders classification problem proposed by de Zeeuw. Our second and third results are sum-product type theorems for two polynomials, generalizing the classical result by Erdős and Szemerédi as well as a theorem by Shen. We also obtain similar results for C, and from this deduce results for fields of characteristic 0 and fields of large prime characteristic. The proofs of our results use tools from semialgebraic/o-minimal geometry
Source Title: Discrete Analysis
ISSN: 2397-3129
DOI: 10.19086/da.55555
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