Semialgebraic methods and generalized sum-product phenomena

Abstract

For a bivariate $P(x,y)\in \mathbb{R}[x,y]\setminus (\mathbb{R}[x]\cup \mathbb{R}[y])$, our first result shows that for all finite $A\subseteq \mathbb{R}$, $|P(A,A)|\ge \alpha |A{|}^{5/4}$ with $\alpha =\alpha (\mathrm{deg}P)\in {\mathbb{R}}^{>0}$ unless $$P(x,y)=f(\gamma u(x)+\delta u(y))\text{ or }P(x,y)=f(u^m(x)u^n(y))$$ for some univariate $f,u\in \mathbb{R}[t]\setminus \mathbb{R}$, constants $\gamma ,\delta \in {\mathbb{R}}^{\ne 0}$, and $m,n\in {\mathbb{N}}^{\ge 1}$. 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 Erdos and Szemerédi as well as a theorem by Shen. We also obtained similar results for $\mathbb{C}$, 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.

23 pages

References in corpus (1)

• Elekes-Rónyai Theorem revisited

Cited by in corpus (1)

• On the dimension of exceptional parameters for nonlinear projections, and the discretized Elekes-Rónyai theorem