Assumptions for uniqueness of analytic functions

Stream: maths

[Yury G. Kudryashov]

In docs#AnalyticOnNhd.eqOn_zero_of_preconnected_of_eventuallyEq_zero, we assume that the function is analytic on a neighborhood of a set. For some sets, it's enough to assume that the function is analytic on the set itself. AFAICT, this includes

• sets that are subsets of the closures of their interior
• all preconnected sets in $\mathbb C$

but this doesn't include all sets with docs#UniqueDiffOn property (I don't have a counterexample right away, but one of the steps of the proof doesn't work for the union of two axes in $\mathbb C^2$). One sufficient condition is that the tangent cone at each point has an innter point. Is this the right assumption to make? Or should I define UniqueAnalyticOn as "a set for which docs#AnalyticOnNhd.eqOn_zero_of_preconnected_of_eventuallyEq_zero holds with codomain = base field", then prove theorems like IsOpen.uniqueAnalyticOn, UniqueAnalyticOn.of_subset_closure etc?

[Yury G. Kudryashov]

#xy: For functions on a docs#Metric.closedBall, I want to assume AnalyticOn, not AnalyticOnNhd.

[Sébastien Gouëzel]

For functions on a closed ball, shouldn't you only assume analytic in the open ball, continuous on the whole ball? That's a much more natural assumption (because it shows up often) and it's enough for the uniqueness, right?

[Yury G. Kudryashov]

Indeed, I can add a version assuming docs#DiffContOnCl for any open set (incl. open balls).

[Yury G. Kudryashov]

Do we want to have a version that works for any connected set in $\mathbb C$?

[Sébastien Gouëzel]

Yury G. Kudryashov said:

Indeed, I can add a version assuming docs#DiffContOnCl for any open set (incl. open balls).

Isn't it more subtle than that? If you take two open balls that are tangent at a point, and their closure, then you have analytic functions on the open set, continuous on its (connected) closure, which coincide locally but not globally.
It's ok if you assume that the open set is connected, though.

[Yury G. Kudryashov]

You're right, I need to assume that the open set is connected.

[Antoine Chambert-Loir]

Yury G. Kudryashov said:

One sufficient condition is that the tangent cone at each point has an inner point. Is this the right assumption to make?

This would be a possible assumption, although, under the assumption of connectedness, I believe that one single such point would suffice. And probably even more general stuff regarding the set of possible directions of the tangent cone.
(I can't find something new in dimension 2, but in dimension 3, one can probably iterate.)

The tautological assumption is density for the analytic Zariski topology, but that's probably difficult to describe explicitely.

Why not introducing a predicate on sets that people can populate with as many relevant examples than needed?