Conventions on this page, and what the several-variable identity theorem does not say

Coordinates and multi-indices on this page are indexed from $0$, exactly as on the library's Euclidean pages. Thus a point of ${\mathbb{C}}^m$ is $z=(z_0,\dots ,z_{m-1})$, a multi-index is $\alpha =({\alpha }_0,\dots ,{\alpha }_{m-1})$ ($C^k$ maps and multi-index derivative notation in Euclidean space), and the polydisc notation is coordinatewise (Balls, polydiscs and the distinguished boundary in ${\mathbb{C}}^m$).

Holomorphic means complex differentiable, and nothing more. Holomorphic functions on an open subset of ${\mathbb{C}}^m$ does not build in continuity, local boundedness or power-series representability. That is why Osgood's lemma: continuous and separately holomorphic implies holomorphic and Locally bounded and separately holomorphic implies holomorphic are theorems rather than tautologies, and why Multi-indexed power series in ${\mathbb{C}}^m$ and their absolute convergence is a separate object rather than the definition of holomorphy.

The distinguished boundary is the one that carries the Cauchy theory here. The polydisc Cauchy formula is an iterated sequence of one-variable contour integrals over the coordinate circles whose product is the distinguished boundary; it does not define an integral over that boundary as a set. The power-series coefficients and Cauchy estimates use those same circles, rather than the whole topological boundary of a closed polydisc. The companion example A function whose modulus attains its maximum only on the distinguished boundary of a bidisc ↗ shows exactly why: the topological boundary contains points where the modulus of a holomorphic function can be far from maximal.

The several-variable identity theorem is weaker than the one-variable one. A holomorphic function vanishing on a nonempty open subset of a domain vanishes identically assumes a nonempty open set of zeros. It does not say that an accumulation point of the zero set is enough, and that stronger statement is false in several variables: the companion false statement A holomorphic function on a domain in ${\mathbb{C}}^2$ vanishing on a set with an accumulation point vanishes identically ↗ records the witness. The gap is structural, not cosmetic. In one variable, a nonzero holomorphic function has isolated zeros; in several variables a zero set may contain whole positive-dimensional complex pieces.