Valuation mathematics

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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.

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: KG ∪ {∞}

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.

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).

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 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 sK can then be written as

s = q/r · p k

where q, rR 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.

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

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 fR

ν(f) = kf ∈ μk but f ∉ μk + 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.

For example, take as R 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 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:

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 numbers, K = C(x,y) and R the ring of polynomials R = C[x, y], and consider the power series

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f = y - \sum_{n=3}^{\infty} \frac{x^n}{n!}}

whose zeros can be parametrized as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (f = 0) \Leftrightarrow x = t, y = \sum_{n=3}^{\infty}t^i.}

Define, for any P(x, y) ∈ R,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nu(P) = \rm{ord}_{t}(P(t,\sum_{n=3}^{\infty}t^i)),}

(the order in t after substituting x and y for their series in (f = 0)), and for P/QK, 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,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nu(x^6 - y^2) = \rm{ord}_t(t^6 - t^6-2t^7-3t^8-\dots )= \rm{ord}_t (-2t^7-3t^8-\dots )=7,}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nu(x) = \rm{ord}_t(t) = 1,}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nu\left(\frac{x^6 - y^2}{x}\right)= 7 - 1 = 6.}

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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