Isolated Zeros

Stream: maths

[Daniel Weber]

In docs#AnalyticOn.eqOn_zero_of_preconnected_of_eventuallyEq_zero_aux, is it possible to weaken the f =ᶠ[𝓝 z₀] 0 assumption to f =ᶠ[𝓝[U] z₀] 0? It sounds to me like it should be possible, but I'm not sure how to prove it. I'm not sure if the same proof can work, as {x | f =ᶠ[𝓝[U] x] 0} might not be open. Any suggestions for how to prove it?

[Sébastien Gouëzel]

That's not true, I think. In $\mathbb{C}^2$, take for U the union of $\mathbb{C} \times \{0\}$ and $\{1\} \times \mathbb{C}$, and let f (z, z') = z'. Then, on a neighborhood of 0 in U, the function f vanishes, but still it does not vanish on the whole set U.

[Daniel Weber]

Thanks. It is true in the one-dimensional case, right?

[Daniel Weber]

Suppose we have a non-zero analytic function on a connected set U. What are the possible values for {x ∈ U | f x ≠ 0}? I thought it was docs#Filter.codiscreteWithin of U, but this example shows that's wrong

[Daniel Weber]

(deleted)

[Sébastien Gouëzel]

The zero set should locally be a subvariety (of positive codimension) and I don't think you can really go beyond that.

[Daniel Weber]

I apologize, I don't know algebraic geometry, could you clarify?

[Sébastien Gouëzel]

In dimension 2, the zero set should be a union of curves and points.

[Antoine Chambert-Loir]

The principle of isolated zeroes requires not only connectedness but also irreducibility: the set should not be the union of two smaller analytic subsets.
For the probably simplest possible example, in 2 variables, take $f(z,w)=z$ on the set defined by the equation $zw=0$.

[Vincent Beffara]

In dimension 1 you might be looking for docs#AnalyticOn.eqOn_zero_of_preconnected_of_frequently_eq_zero (plus some glue saying that a connected set that is not a point has no isolated point so f =ᶠ[𝓝[U] z₀] 0 implies that it is frequently zero)

[Daniel Weber]

That glue is docs#IsPreconnected.preperfect_of_nontrivial, right? I want to add the result that an analytic function which is frequently zero along codiscreteWithin is zero everywhere, so I'm trying to figure out the right generality. Is it simply one dimension?