Weierstrass preparation theorem

The Weierstrass preparation theorem factors a convergent power series regular in one variable as a unique product of a unit and a monic polynomial in that variable.

Weierstrass division theorem: A theorem that divides a convergent power series by a distinguished polynomial, leaving a unique remainder of lower degree. Division by the prepared polynomial yields the preparation theorem and explains its uniqueness.
Convergent power series: A power series that converges within some neighborhood of its center. The theorem concerns analytic germs represented by such series.
Analytic set: A subset locally defined as the common zero set of holomorphic functions. Preparation reduces local equations to polynomial equations in one variable.
Karl Weierstrass: A German mathematician whose work helped establish modern function theory and rigorous analysis. The theorem bears his name and reflects his systematic study of analytic functions.
Hensel's lemma: A lemma lifting factorizations or solutions from a residue field to a complete local ring. It is an algebraic lifting principle, while preparation directly factors an analytic germ.
Weierstrass polynomial: A monic polynomial in one variable whose lower-degree coefficients vanish at the origin. This is the finite polynomial factor produced by preparation.
Formal power series: An algebraic power series manipulated coefficient by coefficient without requiring convergence. The theorem has formal analogues, but its stated analytic form requires convergence.
Local algebra: The study of algebraic structures localized near a point or prime ideal. The theorem transfers analytic germ problems into finite local algebra.
Weierstrass factorization theorem: A theorem constructing entire functions with prescribed zeros and multiplicities. It belongs to Weierstrass's broader program of representing analytic functions through their zeros.
Formal preparation theorem: The formal power-series analogue of Weierstrass preparation, factoring a regular series into a unit and distinguished polynomial. It replaces analytic convergence with formal algebraic power series.

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Hartogs extension theoremRelated: It became a central tool for analyzing local holomorphic structure in the subject Hartogs helped establish.