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https://doi.org/10.19086/da.55555
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. https://doi.org/10.19086/da.55555 | 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 | URI: | https://scholarbank.nus.edu.sg/handle/10635/242963 | ISSN: | 2397-3129 | DOI: | 10.19086/da.55555 |
Appears in Collections: | Elements Staff Publications |
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