Valuation (mathematics)

Algebra and algebraic geometry

In algebra (or algebraic geometry), valuations are, in some sense, the generalization to commutative algebra of the geometrical concept of contact between two algebraic or analytic varieties.

Related Topics:
Algebra - Algebraic geometry - Commutative algebra - Contact - Algebraic - Analytic

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Given a field K and a commutative ordered group (G, + , >), a valuation is a map

Related Topics:
Field - Commutative - Ordered group - Map

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:ν: K → G ∪ {∞}

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where ∞ is a symbol with the property that ∞ ≥ g for any g ∈ G) satisfying the following conditions:

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• ν(0) = ∞. The geometrical translation of this is that any non-empty germ of variety near a point contains that point.
• ν(ab) = ν(a) + ν(b) for any a, b in K^∗. This is the same as saying that ν is a group homomorphism between K^∗ and G.
• ν(a + b) ≥ min(ν(a), ν(b)). In some sense, this is a translation of the triangle inequality of metric spaces.

Two valuations are said to be equivalent if they are proportional (i.e. they differ by a fixed element in G). An equivalence class of valuations on a field K is called a place. Ostrowski's theorem completely classifies the places of the field Q (these correspond to the equivalence classes of valuations for the real and p-adic completions).

Related Topics:
Ostrowski's theorem - Real - P-adic - Completions

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Usually (and we are going to do it in the sequel), ν is required to be surjective, especially because many arguments are done using preimages of elements of G.

Related Topics:
Surjective - Preimages

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Examples

Example 1. Let K be the quotient field of a principal ideal domain R. Let p be any irreducible element of R so that the ideal (p) is prime. Any element g of R belongs to some power (p)k, k ≥ 0, of the ideal (p). If g = 0, it belongs to (p)k for any k, while if g is coprime with p, we let k = 0. Any nonzero element s ∈ K can then be written as

Related Topics:
Quotient field - Principal ideal domain - Irreducible element - Coprime

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:s = q/r · p k

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where q, r ∈ R are coprime with p, and k is an integer. Defining ν(s) = k and ν(0) = ∞ gives a valuation from K to Z, the additive group of integers.

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When R is Z and p is a prime number, this called the p-adic valuation on the rational numbers.

Related Topics:
Prime number - Rational number

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Example 2. Let (R, μ) be a local integral ring with maximal ideal μ. Any f in R belongs to some power k of μ. Define, for any f ∈ R

Related Topics:
Local integral ring - Maximal ideal

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:ν(f) = k ⇔ f ∈ μk but f ∉ μk + 1

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and extend it to the quotient field K of R as follows:

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:ν(f/g) = ν(f) − ν(g);

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this is easily proved to be well-defined. Also, ν(0) = ∞ as usual. This is the μ-adic valuation on K.

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For example, take as R the ring of formal power series over a field. To be more specific, let R be Cx, y the ring of formal power series in two variables over the complex numbers and μ = (x, y) its maximal ideal. The μ-adic valuation in this case is given by the difference of the orders of the power series in the numerator and the denominator:

Related Topics:
Formal power series - ''x'', ''y'' - Orders

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:ν(x2 + y2 + x3y2) = 2

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:ν(x3/y2)= 3 − 2 = 1

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Example 3. (Geometrical notion of contact). For simplicity, let K be the field of rational functions in two variables over the complex numbers, K = C(x,y) and R the ring of polynomials R = C, and consider the power series

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:f = y - sum_{n=3}^{infty} rac{x^n}{n!}

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whose zeros can be parametrized as

Related Topics:
Zeros - Parametrized

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:(f = 0) Leftrightarrow x = t, y = sum_{n=3}^{infty}t^i.

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Define, for any P(x, y) ∈ R,

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: u(P) = m{ord}_{t}(P(t,sum_{n=3}^{infty}t^i)),

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(the order in t after substituting x and y for their series in (f = 0)), and for P/Q ∈ K∗, put ν(P/Q) = ν(P) − ν(Q). As the power series defining f is non-polynomial, it is easy to prove that this ν is a valuation, and ν(P) is the intersection number between the curves (P = 0) and (f = 0). Specifically,

Related Topics:
Intersection number - Curves

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: u(x^6 - y^2) = m{ord}_t(t^6 - t^6-2t^7-3t^8-dots )= m{ord}_t (-2t^7-3t^8-dots )=7,

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: u(x) = m{ord}_t(t) = 1,

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: uleft(rac{x^6 - y^2}{x} ight)= 7 - 1 = 6.

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All the examples are of Dedekind valuations, which are those for which G is the additive group of the integers (Z, +).

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