Semialgebraic methods and generalized sum-product phenomena

For a bivariate $P(x,y) \in $\mathbb{R}$[x,y]\setminus ($\mathbb{R}$[x] \cup $\mathbb{R}$[y])$,ourfirstresultshowsthatforallfinite$A \subseteq $\mathbb{R}$$,$|P(A,A)|\geq \alpha|A|^{5/4}$with$\alpha =\alpha($\mathrm{d}\mathrm{e}\mathrm{g}$ P) \in $\mathbb{R}$^{>0}$unless$$ P(x,y)=f(\gamma u(x)+\delta u(y)) $\text{ or }$ P(x,y)=f(u^m(x)u^n(y)) $$forsomeunivariate$f, u \in $\mathbb{R}$[t]\setminus $\mathbb{R}$$,constants$\gamma, \delta \in $\mathbb{R}$^{\neq 0}$,and$m, n\in $\mathbb{N}$^{\geq 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 Szemer'edi 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.