Examples of irreducible holomorphic function in more than one variable.

Mathematics Asked by Alain Ngalani on November 27, 2020

I’m studying analysis in several complex variables and in particular Weierstrass preparation theorem caught my interest (I’ll include the theorem for clarity).

In the examples I came up with I only found functions that could be written as the product of an unit and linear Weierstrass polynomials.

It’s never implied that every Weierstrass irreducible polynomial is of degree 1 or that in general irreducible functions are the degree 1 polynomial so I was looking for some examples of holomoprhic functions that are irreducible (at the origin) but are not polynomials of degree 1.enter image description here

One Answer

I actaully don't remember about details on why this is but I've been told in several occasions that polynomials like $z^2-xy^2$ are irreducible. I hope this helps

Correct answer by Frankie123 on November 27, 2020

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