Communication complexity studies how much information must be exchanged to solve a problem whose input is split among two or more parties. In this work, we study an algebraic variant, where the inputs are vectors over a field $\mathbb{F} \in \{\mathbb{R}, \mathbb{C}\}$. There are two players, Alice and Bob, with inputs $X\in \mathbb{F}^n$ and $Y\in \mathbb{F}^n$, respectively. In the polynomial evaluation problem, the goal is to compute the value of a polynomial $g\in \mathbb{F}[X,Y]$ on the given input. In the set-recognition problem, the goal is to decide whether $(X,Y)\in S$, for a given set $S\subseteq \mathbb{F}^{n} \times \mathbb{F}^n$. In both settings, Alice and Bob send evaluations of polynomials depending only on their own inputs. In the set-recognition problem, a referee receives the messages and may apply polynomial tests to the messages received so far; the outcomes of these tests determine acceptance or rejection. The protocols may be deterministic or probabilistic. We give non-trivial upper bounds for a range of natural polynomial evaluation and set-recognition problems. We also prove reductions between different problems. Our main technical contribution is a general framework for proving lower bounds for algebraic set-recognition problems. Using this framework, we prove several probabilistic lower bounds for natural problems. In a number of cases, these lower bounds match the corresponding upper bounds, giving tight or near-tight characterizations of their algebraic communication. Finally, we give two applications of our framework. First, we use the communication lower bounds to prove lower bounds for a class of left-to-right algebraic algorithms, which we call algebraic scanners. Second, we show that our lower bound arguments extend beyond polynomials to a more general algebraic computational setting inspired by the Blum--Shub--Smale model.

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