Math 2.0

Commutative algebra Completion and topological rings Theorem

Artin approximation theorem

Builds on Multivariate formal power seriesPolynomial ringHenselizationI-adic topology

Statement

Let kk be a field and let x=(x1,…,xn)x = (x_1, \ldots, x_n) be variables. Let k[ ⁣[x] ⁣]k[\![x]\!] be the ring of formal power series in xx. Let k⟨x⟩⊆k[ ⁣[x] ⁣]k\langle x \rangle \subseteq k[\![x]\!] be the ring of algebraic power series: those that are algebraic over the polynomial ring k[x]k[x]. Let y=(y1,…,yN)y = (y_1, \ldots, y_N) be further variables and let f1,…,fm∈k[x,y]f_1, \ldots, f_m \in k[x, y] be polynomials. Assume that y^=(y^1,…,y^N)∈k[ ⁣[x] ⁣]N\hat{y} = (\hat{y}_1, \ldots, \hat{y}_N) \in k[\![x]\!]^N satisfies fj(x,y^)=0f_j(x, \hat{y}) = 0 for j=1,…,mj = 1, \ldots, m. Then for every integer c≥1c \ge 1 there is yˉ∈k⟨x⟩N\bar{y} \in k\langle x \rangle^N such that fj(x,yˉ)=0f_j(x, \bar{y}) = 0 for j=1,…,mj = 1, \ldots, m and yˉi≡y^i(mod(x1,…,xn)c)\bar{y}_i \equiv \hat{y}_i \pmod{(x_1, \ldots, x_n)^c} for i=1,…,Ni = 1, \ldots, N.

Remarks

The ring k⟨x⟩k\langle x \rangle is the Henselization of the local ring of k[x]k[x] at the ideal (x1,…,xn)(x_1, \ldots, x_n). The theorem says that formal solutions of polynomial equations can be approximated, in the (x)(x)-adic topology, by algebraic solutions. M. Artin proved it in 1969. Popescu's theorem extends the result to every excellent Henselian local ring in place of k⟨x⟩k\langle x \rangle.

Example: the equation y1−y2=0y_1 - y_2 = 0 has the formal solution y^1=y^2=∑i≥0xi/i!\hat{y}_1 = \hat{y}_2 = \sum_{i \ge 0} x^i / i! over ℚ\Q, which is not algebraic. The truncations yˉ1=yˉ2=∑i<cxi/i!\bar{y}_1 = \bar{y}_2 = \sum_{i < c} x^i / i! are polynomials, hence algebraic, and agree with y^\hat{y} modulo (x)c(x)^c. The theorem does not say that the formal solution itself is algebraic; it only gives algebraic solutions that agree with it to any given order.

Going deeper

For a student

In algebraic geometry, formal power series describe infinitesimal neighbourhoods, while algebraic power series describe étale neighbourhoods. The theorem lets one pass from a formal statement about the completion of a local ring to a statement in the Henselization, and so to an étale neighbourhood. This is used, for example, to compare the formal and the étale local structure of singularities. It is also a key input in Artin's work on algebraic spaces and moduli problems.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

Show the path as a graph
Henselization, concept, no evidence levelHenselizationconceptI-adic topology, concept, no evidence levelI-adic topologyconceptMultivariate formal power series, concept, no evidence levelMultivariate formal p…conceptPolynomial ring, concept, no evidence levelPolynomial ringconceptArtin approximation theorem, statement, no evidence levelArtin approximation t…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Henselization concept
  2. I-adic topology concept
  3. Multivariate formal power series concept
  4. Polynomial ring concept
  5. Artin approximation theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Artin_approximation_theorem
About this page
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Kindstatement, theorem
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MSC13B40, 13J15, 13F25
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AboutMultivariate formal power series, Henselization
UsesMultivariate formal power series, Polynomial ring, Henselization, I-adic topology
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Changed2 November 2026

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