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Informally, a valuation is an assignment of particular values to the variables in a mathematical statement or equation.

In logic and model theory, a valuation is a map from the set of variables of a language of order 0 or 1 to the universe of some interpretation of that language.

In algebra (in particular in algebraic geometry or algebraic number theory), valuations are a measure of size or multiplicity. They generalize to commutative algebra the notion of size inherent in consideration of the degree of a pole or multiplicity of a zero in complex analysis, the degree divisibility of a number by a prime number in number theory, and the geometrical concept of contact between two algebraic or analytic varieties in algebraic geometry.

Contents 1 Logic/Model theory definition 2 Algebraic definition 3 Examples 3.1 Logical equality 3.2 <math>p</math>-adic valuation 3.3 <math>\mu</math>-adic valuation 3.4 Geometric notion of contact 4 See also 5 Rererences 6 External links

[edit] Logic/Model theory definition

First of all, consider a formal language <math>\boldsymbol{L}=\{\boldsymbol{A},\boldsymbol{F}\}</math>, where <math>\boldsymbol{A}</math> is its alphabet, the set <math> \boldsymbol{V_0} \in \boldsymbol{A_0}</math> is its propositional variables set, and the set <math>\boldsymbol{F}</math> is its set of formulas. let <math>\mathfrak{A}</math> be a abstract algebra with tree binary operations and one unary operation: for languages of order 0 and 1 it can be the algebra of formulas <math>\{\boldsymbol{F},\vee,\wedge,\Rightarrow,-\}</math> with properly defined logical disjunction <math>\vee</math>, logical conjunction <math>\wedge</math>, logical implication <math>\Rightarrow</math> and logical negation <math>-\,</math>. Given this mathematical objects, a valuation is a any point of the cartesian product <math>\mathfrak{A}^\boldsymbol{V_0}</math>, that is any map

<math>v:\boldsymbol{V_0}\rightarrow\mathfrak{A}</math>

which maps propositional variables to algebraic formulas in <math>\mathfrak{A}</math>. Details on those deep logic concepts can be found in the second reference listed in the reference section.

[edit] Algebraic definition

Consider a field <math>\mathbbTemplate:K</math> and a commutative ordered group <math> (\mathfrakTemplate:G,+,\geq) </math>, and adjoin to <math>\mathfrakTemplate:G</math> an element <math>\infty </math> such that

• <math>\infty \geq \mathfrakTemplate:A\quad\forall\mathfrakTemplate:A\in\mathfrak{G}\cup\{\infty\}</math>
• <math>\infty + \infty =\infty</math> in <math>\mathfrak{G}\cup\{\infty\}</math>
• <math>\infty + \mathfrak{a} = \mathfrak{a} + \infty = \infty\quad\forall\mathfrak{a}\in\mathfrak{G}\cup\{\infty\}</math>

A (exponential) valuation is then a map <math> v:\mathbb{K} \rightarrow \mathfrak{G} </math> satisfying the following properties

• <math> v(a)=\infty\iff{a=0}</math> (in geometric applications, this property means that any non-empty germ of a analytic variety near a point contains that point)
• <math>v(ab)=v(a)+v(b)\,</math> for all <math>a,b\in\mathbbTemplate:K^*</math> (equivalently <math>v\,</math> is a group homomorphism between <math>\mathbbTemplate:K^*\,</math> and <math>\mathfrakTemplate:G</math>)
• <math>v(a+b)\geq\mathrm{min}\{v(a),v(b)\}</math> for all <math>a,b\in\mathbbTemplate:K</math> (this property is a translation of the triangle inequality of metric spaces to ordered groups)

where <math>\mathbbTemplate:K^*</math> is the multiplicative subgroup of the given field. There is also a dual definition of the same object, involving an element <math>\mathfrak{0}</math> such that

• <math>\mathfrak{0} \leq \mathfrakTemplate:A\quad\forall\mathfrakTemplate:A\in\mathfrak{G}\cup\{\mathfrak{0}\}</math>
• <math>\mathfrak{0}\mathfrak{0} =\mathfrak{0}</math> in <math>\mathfrak{G}\cup\{\infty\}</math>
• <math>\mathfrak{0}\mathfrak{a} = \mathfrak{a}\mathfrak{0} = \mathfrak{a}\quad\forall\mathfrak{a}\in\mathfrak{G}\cup\{\infty\}</math>

Following this line of thought, a valuation is then defined as a map <math> v:\mathbb{K} \rightarrow \mathfrak{G} </math> satisfying the following properties

• <math> v(a)=\mathfrak{0}\iff{a=0}</math>
• <math>v(ab)=v(a)v(b)\,</math> for all <math>a,b\in\mathbbTemplate:K^*</math>
• <math>v(a+b)\leq\mathrm{max}\{v(a),v(b)\}</math> for all <math>a,b\in\mathbbTemplate:K</math>

The definition involving the element <math>\infty</math> is more frequently used in ordinary mathematical research: in the following considerations this is the only definition used.

Two valuations are said to be equivalent if they are proportional (i.e. they differ by a fixed element in <math>\mathfrakTemplate:G</math>): every equivalence class of valuations on a field <math>\mathbbTemplate:K</math> is called a place. Ostrowski's theorem completely classifies places of every field <math>\mathbbTemplate:K</math>: these are precisely equivalence classes of valuations for the real and p-adic completions) of the chosen field <math>\mathbbTemplate:K</math>.

Usually a valutation <math>v:\mathbb{K}\rightarrow\mathfrak{G}\cup\{\infty\}</math> is required to be surjective, since many arguments used in ordinary mathematical research involving those objects use preimages of unspecified elements of <math>\mathfrakTemplate:G</math>.

[edit] Examples

All the following examples, except the first one, deal with Dedekind valuations, that is valuations for which the ordered abelian group <math> \mathfrak{G} </math> is the additive group of the integers, i.e. <math>\mathfrak{G}=(\mathbb{Z},+)</math>: all shown valuations are surjective.

[edit] Logical equality

A very simple example of valuation, illustrating a basic part of the process of formalization of logical arguments using mathematical symbols, is the following: the statement

"<math>x = y\,</math>"

is satisfied by (i.e. true for) every valuations in which "<math>x\,</math>" is mapped to the same value as "<math>y\,</math>", and not satisfied by (i.e. false for) all other valuations.

[edit] <math>p</math>-adic valuation

Let <math>\mathbbTemplate:K</math> be the field of fractions of a principal ideal domain <math>\mathfrak</math>, and let <math> \mathfrak \in \mathfrak </math> be one of its irreducible elements. If ideal <math>(\mathfrak{p})\,</math> is prime, then any element <math>\mathfrakTemplate:G\in R</math> belongs to some power <math>(\mathfrak)^k\,</math>, for a <math> k \in \mathbb{Z} </math>: if <math>\mathfrakTemplate:G = 0</math>, then <math> \mathfrak{g} \in (\mathfrak{p})^k</math> for any <math>k\,</math>, while if <math> \mathfrak{g} </math> is coprime respect to <math> \mathfrak{p} </math>, it is sufficent to put <math> k = 0 \,</math>. Any element <math> s \in \mathbb{K} </math> can then be written as

<math> s = \mathfrak{q}/\mathfrak{r} \cdot \mathfrak{p}^k </math>

where <math> \mathfrak{q,r\in R} </math> are coprime respect to <math> \mathfrak{p}</math>, and <math>k\,</math> is an integer. Then, defining the map <math>v:\mathbb{K} \rightarrow \mathbb{Z} </math> as

<math>v(s)=k \quad \forall s \in \mathbb{K}^* </math>

<math>v(0) = \infty \quad 0 \in \mathbb{K} </math>

it is eay to prove that it is a valuation from <math>\mathbbTemplate:K</math> to <math>\mathbbTemplate:Z</math>, the additive group of integers. When <math> \mathfrak{R} = \mathbb{Z} </math> and <math>\mathfrak{p}\equiv p </math> is a prime number, this valuation is called the <math>p\,</math>-adic valuation on the set <math> \mathbb{Q} </math> of rational numbers.

[edit] <math>\mu</math>-adic valuation

Let <math>(\mathfrak,\mu)</math> be a local integral ring with maximal ideal <math>\mu\,</math>: since any <math>\mathfrak

• [[{{{1}}} in film|{{{1}}}]] - [[{{{2}}}]]\in\mathfrak</math> belongs to some integer power <math> k\in\mathbb{N} </math> of the ideal <math>\mu\,</math>, define the map <math>v:\mathfrak{R}\rightarrow\mathbb{Z}</math> as

<math> v(\mathfrak{f}) = k\,\Longleftrightarrow\,\mathfrak

• [[{{{1}}} in film|{{{1}}}]] - [[{{{2}}}]]\in\mu^k </math> and <math> \mathfrak
• [[{{{1}}} in film|{{{1}}}]] - [[{{{2}}}]]\notin\mu^Template:K+ 1 </math> for all <math> \mathfrak
• [[{{{1}}} in film|{{{1}}}]] - [[{{{2}}}]] \in \mathfrak </math>

and extend it to the field of fractions <math>\mathbbTemplate:K</math> of <math>\mathfrak</math> as follows:

<math> v\mathfrakTemplate:(f/g) = v(\mathfrak

• [[{{{1}}} in film|{{{1}}}]] - [[{{{2}}}]]) - v(\mathfrakTemplate:G) \quad \forall \mathfrak{f/g} \in \mathbb{K}^* </math>

<math> v(\mathfrak0) = \infty \quad \mathfrak{0} \in \mathbb{K} </math>

It is easy to prove that this map is a well-defined valuation: it is called <math>\mu</math>-adic valuation on <math>\mathbb{K}</math>. If, for example, <math>\mathfrak{R}</math> is the ring of formal power series in two variables over the complex field <math>\mathbb{C}</math>, i.e. <math>\mathfrak{R} = \mathbb{C}x,y</math> and <math>\mu = (x,y)\,</math> is its maximal ideal, its <math>\,\mu</math>-adic valuation is given by the difference of the orders of the power series in the numerator and the denominator: as examples, computation of <math>\mu</math>-valuation for some fractions follows

<math> v(x^2 + y^2 + x^3y^2)=2 \,</math>

<math> v(x^3/y^2) = 3 - 2 = 1 \,</math>

[edit] Geometric notion of contact

Let <math> \mathbbTemplate:K = \mathbb[[:Category:{{{1}}}|{{{1}}}]](x,y) </math> be the field of rational functions of two variables over the complex field, <math> \mathfrak = \mathbb[[:Category:{{{1}}}|{{{1}}}]][x,y] </math> the ring of polynomials over the same field, and consider the power series

<math> f(x,y) = y - \sum_{n=3}^{\infty} \frac{x^n}{n!} </math>

whose zero set, the analytic variety <math>V_f\,</math> can be parametrized by one coordinate <math>t\,</math> as follows

<math> V_f = \{(x,y)\in\mathbb{C}^2\,|\, f(x,y) = 0\} = \left\{ (x,y)\in\mathbb{C}^2\,|\,(x,y) = \left(t,\sum_{n=3}^{\infty}t^i\right)\right\}</math>

For any <math>P(x,y)\in\mathfrak{R}</math>it is possible to define the map <math>v:\mathfrak{R}\rightarrow\mathbb{Z}</math> as the value of the order of the formal power series in the variable <math>t\,</math> obtained by calculating <math>P\,</math> on any point of <math>V_f\,</math>

<math> v(P) = \mathrm{ord}(P)|_{V_f} = {\mathrm{ord}}_t \left(P\left(t,\sum_{n=3}^{\infty}t^i\right)\right),</math>

It is possible to extend the map <math>v\,</math> from its original domain of definition to the field <math>\mathbb{K}</math> as follows

<math> v(P/Q) = v(P) - v(Q)\quad\forall P/Q \in \mathbb{K}^* </math>

<math> v(0)=\infty \quad 0 \in \mathbb{K} </math>

As the power series <math>f\,</math> is not a polynomial, it is easy to prove that the extended map <math>v\,</math> is a valuation: <math>v(P)\,</math> is called intersection number between the curves (1-dimensional analytic varieties) <math>V_P\,</math> and <math>V_f\,</math>. As an example, the computation of some intersection numbers follows

<math>v(x) = \mathrm{ord}_t(t) = 1\,</math>

<math>v(x^6-y^2)=\mathrm{ord}_t(t^6-t^6-2t^7-3t^8-\dots)=\mathrm{ord}_t (-2t^7-3t^8-\dots)=7</math>

<math>v\left(\frac{x^6 - y^2}{x}\right)= 7 - 1 = 6</math>

[edit] See also

• Local ring
• Valuation ring
• Absolute value (algebra)

[edit] Rererences

• Nathan Jacobson (1989). Basic algebra II, 2^nd ed., W H Freeman. ISBN 0-7167-1933-9., chapter 9 paragraph 6 Valuations.
• Helena Rasiowa and Roman Sikorski (1970). The Mathematics of Metamathematics, 3^rd ed., PWN., chapter 6 Algebra of formalized languages.

[edit] External links

• Valuation, Planetmath.org Encyclopedia [1].it:Valutazione

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