Semialgebraic methods and generalized sum-product phenomena

Yifan Jing, Souktik Roy, Chieu-Minh Tran

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)|\geq α|A|^{5…