Valuation (mathematics)

Table of contents

1 Model Theory 2 Algebra and Algebraic Geometry 3 Examples

Model Theory

In logic and model theory, a valuation is a map from the set of variables of a first-order language to the universe of some interpretation of that language.

Informally, it is an assignment of particular values to the variables in a mathematical statement or equation. So for example the statement "x=y" is satisfied by (i.e. true for) valuations in which "x" is mapped to the same value as "y", and not satisfied by (i.e. false for) all other valuations. This may seem trivial in such a simple case, but is part of the process of formalising logical arguments using mathematical symbols.

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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.

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

ν: K → G ∪ {∞}

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

1. ν(0) = ∞. The geometrical translation of this is that any non-empty germ of variety near a point contains that point.

2. 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.
3. ν(a+b) ≥ min(ν(a),ν(b)). In some sense, this is a translation of the triangle inequality of metric spaces.

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.

Examples

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

s=p/q . f^k

where p, q ∈ R are coprime with f and k is an integer. Defining ν(s)=k (and ν(0)=∞) gives a valuation from K to Z (the additive group of integer numbers).
The above, when R is Z (the integers) and p is a prime number is called the p-adic valuation over Q (the rational numbers, which is the quotient field of Z).

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

ν(f) = k ⇔ f ∈ μ^k but f ∉ μ^k+1

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

ν(f/g) = ν(f)- ν(g)

(this is easily proved to be well-defined). Also, ν(0)=∞ as usual. This is the μ-adic valuation on K.
An instance of this example is given by R being the ring of formal power series over a field. To be more specific, let R be C[[x,y]] the ring of formal power series in 2 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:

ν(x²+y²+x³y²) = 2

ν(x³/y²)= 3-2 = 1

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

whose zeros can be parametrized as

Define, for any P(x,y) ∈ R,

(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,

Etc...

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