An algebraic circuit over a commutative ring $\mathbb{K}$ with variables from a set $V$ is a finite directed acyclic graph whose input gates are labelled by variables or constants of $\mathbb{K}$ and whose internal gates are labelled by $+$ or $\times$. We allow unbounded fan-in and parallel wires: the number of wires from a gate $h$ to a gate $g$ is the multiplicity $m(g,h) \in \mathbb{N}$, and $h$ is a child of $g$ if $m(g,h) > 0$. Input gates have no children. There is a distinguished output gate, and every gate lies on a path to it.

Every gate $g$ computes a polynomial $p_g \in \mathbb{K}[V]$: an input gate computes its label, and

if $g$ is a sum or a product gate, respectively, so that wires are counted with multiplicity and empty sums and products are $0$ and $1$. The circuit computes the polynomial of its output gate. Formally, "the gate $g$ computes $p$" is defined inductively by these rules, so that exactly the polynomials obtained by evaluating the circuit bottom-up are computed. Every gate computes exactly one polynomial; the values $p_h$ at children $h$ with $m(g,h) = 0$ do not contribute.

The size of a circuit is the number of its gates and wires, counting multiplicities. A path of length $m$ is a sequence of gates $g_0, \dots, g_m$ such that $g_{i+1}$ is a child of $g_i$ for all $i < m$. The depth is the maximum length of a path. Since every gate lies on a path to the output gate, this is the maximum number of wires on a path from the output gate to a gate without children, e.g. an input gate.

A circuit is reduced if no two distinct input gates have the same label and no two distinct internal gates have the same label and the same multiset of children, i.e. the same multiplicities $m(g, \cdot)$.