Complex Analytic and Differential Geometry 2024, 51: Local Rings of Germs of Analytic Functions

For a holomorphic g on a neighborhood of the origin in n-dimensional complex space, with g(0, zₙ)/zʳₙ has a non-zero finite limit at zₙ = 0. g(z) = u(z)P(z', zₙ) with an invertible holomorphic u in a neighborhood of the polydisk at r', and rₙ, and the Weierstrass polynomial P in zₙ with holomorphic coefficients on a neighborhood in n-1 dimensional complex space.

If g(0) vanishes at order m, and v is in ℂⁿ \ {0}, chosen such that g on v is nonzero, then r = m and P vanishes at order m at 0. Every bounded holomorphic function f on the polydisk can be uniquely represented by q, R analytic in the polydisk and a polynomial R of degree smaller or equal to r - 1

The ring of germs of holomorphic functions in complex space O can be identified with the ring of convergent power in a series of complex numbers. It's Noetherian. For P, F in that ring of germs of holomorphic functions on n-1 complex space on zₙ, where P is Weierstrass polynomial so that P divides F in Oₙ, then P divides F in Oₙ₋₁[zₙ]. If it's a factorial ring, Oₙ is entire and every nonzero germ f in Oₙ admits a factorization f into irreducible elements, and the factorization is unique up to invertible elements. If f, g in Oₙ are relatively prime, then the germs fₛ, gₛ at all points s in ℂⁿ near 0 are relatively prime.