Pairs of Definition and Minimal Pairs · minimal pair for every positive 2vKe. H.Ćmiel,P.Szewczyk...

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Pairs of Definition and Minimal Pairs An overview of results by S.K. Khanduja, V. Alexandru, N. Popescu and A. Zaharescu Hanna Ćmiel and Piotr Szewczyk Institute of Mathematics, University of Szczecin Workshop on ’Valuations on rational function fields’ Szczecin, 8.05.2018

Transcript of Pairs of Definition and Minimal Pairs · minimal pair for every positive 2vKe. H.Ćmiel,P.Szewczyk...

Page 1: Pairs of Definition and Minimal Pairs · minimal pair for every positive 2vKe. H.Ćmiel,P.Szewczyk (University of Szczecin) Pairs of Definition and Minimal Pairs Szczecin, 8.05.2018

Pairs of Definition and Minimal PairsAn overview of results by S.K. Khanduja, V. Alexandru,

N. Popescu and A. Zaharescu

Hanna Ćmiel and Piotr Szewczyk

Institute of Mathematics, University of SzczecinWorkshop on ’Valuations on rational function fields’

Szczecin, 8.05.2018

Page 2: Pairs of Definition and Minimal Pairs · minimal pair for every positive 2vKe. H.Ćmiel,P.Szewczyk (University of Szczecin) Pairs of Definition and Minimal Pairs Szczecin, 8.05.2018

Definition (minimal pair)

A pair (α, δ) ∈ K × v K is said to be a minimal pair (more precisely, a(K , v)-minimal pair) if for every β ∈ K we have

v(α− β) ≥ δ ⇒ [K (α) : K ] ≤ [K (β) : K ],

i.e. α has least degree over K in the closed ball

B(α, δ) = {β ∈ K | v(α− β) ≥ δ}.

Example (minimal pair)

Let f (x) ∈ O[x ] be a monic polynomial of degree m ≥ 1 with(fv)(x)

irreducible over Kv and let α be the root of f (x) in K . Then (α, δ) is aminimal pair for every positive δ ∈ vK .

H.Ćmiel,P.Szewczyk (University of Szczecin) Pairs of Definition and Minimal Pairs Szczecin, 8.05.2018 2 / 17

Page 3: Pairs of Definition and Minimal Pairs · minimal pair for every positive 2vKe. H.Ćmiel,P.Szewczyk (University of Szczecin) Pairs of Definition and Minimal Pairs Szczecin, 8.05.2018

Definition (minimal pair)

A pair (α, δ) ∈ K × v K is said to be a minimal pair (more precisely, a(K , v)-minimal pair) if for every β ∈ K we have

v(α− β) ≥ δ ⇒ [K (α) : K ] ≤ [K (β) : K ],

i.e. α has least degree over K in the closed ball

B(α, δ) = {β ∈ K | v(α− β) ≥ δ}.

Example (minimal pair)

Let f (x) ∈ O[x ] be a monic polynomial of degree m ≥ 1 with(fv)(x)

irreducible over Kv and let α be the root of f (x) in K . Then (α, δ) is aminimal pair for every positive δ ∈ vK .

H.Ćmiel,P.Szewczyk (University of Szczecin) Pairs of Definition and Minimal Pairs Szczecin, 8.05.2018 2 / 17

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Valuation given by a minimal pair

Let (α, δ) ∈ K × v K be a (K , v)-minimal pair. The mapping wαδ definedon K (x) associated with this minimal pair is given by

wαδ

( n∑i=0

ci (x − α)i)

= mini

{v(ci ) + iδ

}, ci ∈ K . (1)

It is shown in [1] that wαδ is indeed a valuation on K . By wαδ we willdenote the restriction of wαδ to K .

ExampleFor the (K , v)-minimal pair (0, 0) we acquire the well known Gaussvaluation:

wαδ

( n∑i=0

cixi

)= min

i

{v(ci )}.

H.Ćmiel,P.Szewczyk (University of Szczecin) Pairs of Definition and Minimal Pairs Szczecin, 8.05.2018 3 / 17

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Valuation given by a minimal pair

Let (α, δ) ∈ K × v K be a (K , v)-minimal pair. The mapping wαδ definedon K (x) associated with this minimal pair is given by

wαδ

( n∑i=0

ci (x − α)i)

= mini

{v(ci ) + iδ

}, ci ∈ K . (1)

It is shown in [1] that wαδ is indeed a valuation on K . By wαδ we willdenote the restriction of wαδ to K .

ExampleFor the (K , v)-minimal pair (0, 0) we acquire the well known Gaussvaluation:

wαδ

( n∑i=0

cixi

)= min

i

{v(ci )}.

H.Ćmiel,P.Szewczyk (University of Szczecin) Pairs of Definition and Minimal Pairs Szczecin, 8.05.2018 3 / 17

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Question: what do we know about wαδ? If we have a given valuation w onK (x), when is it given by a minimal pair?

Theorem A, [2]

The valuation wαδ defined by (1) is a residue transcendental extension ofv to K (x). Conversely, for any residue transcendental extension of v toK (x) there exists a minimal pair (α, δ) such that w = wαδ.

Theorem B, [2]

If (α, δ), (β, η) are two (K , v)-minimal pairs then wαδ = wβη if and only ifδ = η and v(α′ − β) ≥ δ for some K -conjugate α′ of α.

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Question: what do we know about wαδ? If we have a given valuation w onK (x), when is it given by a minimal pair?

Theorem A, [2]

The valuation wαδ defined by (1) is a residue transcendental extension ofv to K (x). Conversely, for any residue transcendental extension of v toK (x) there exists a minimal pair (α, δ) such that w = wαδ.

Theorem B, [2]

If (α, δ), (β, η) are two (K , v)-minimal pairs then wαδ = wβη if and only ifδ = η and v(α′ − β) ≥ δ for some K -conjugate α′ of α.

H.Ćmiel,P.Szewczyk (University of Szczecin) Pairs of Definition and Minimal Pairs Szczecin, 8.05.2018 4 / 17

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Question: what do we know about wαδ? If we have a given valuation w onK (x), when is it given by a minimal pair?

Theorem A, [2]

The valuation wαδ defined by (1) is a residue transcendental extension ofv to K (x). Conversely, for any residue transcendental extension of v toK (x) there exists a minimal pair (α, δ) such that w = wαδ.

Theorem B, [2]

If (α, δ), (β, η) are two (K , v)-minimal pairs then wαδ = wβη if and only ifδ = η and v(α′ − β) ≥ δ for some K -conjugate α′ of α.

H.Ćmiel,P.Szewczyk (University of Szczecin) Pairs of Definition and Minimal Pairs Szczecin, 8.05.2018 4 / 17

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Minimal pairs – different approach

Let w be a given extension of v to K (x) and w an extension of w toK (x). Consider the set

w(x − K ) := {w(x − a) | a ∈ K}.

Theorem 1, [3]

w is a residue transcendental extension if and only if:1 v K = w K (x),2 the set w(x − K ) is upper bounded in w K (x),3 w K (x) contains its upper bound.

Let δ be the upper bound of w(x − K ). Then there exists α ∈ K suchthat δ = w(x − α) and thus ([3]) w is a residue transcendental extensionof v defined by (1). The pair (α, δ) is called a pair of definition.

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Minimal pairs – different approach

Let w be a given extension of v to K (x) and w an extension of w toK (x). Consider the set

w(x − K ) := {w(x − a) | a ∈ K}.

Theorem 1, [3]

w is a residue transcendental extension if and only if:1 v K = w K (x),2 the set w(x − K ) is upper bounded in w K (x),3 w K (x) contains its upper bound.

Let δ be the upper bound of w(x − K ). Then there exists α ∈ K suchthat δ = w(x − α) and thus ([3]) w is a residue transcendental extensionof v defined by (1). The pair (α, δ) is called a pair of definition.

H.Ćmiel,P.Szewczyk (University of Szczecin) Pairs of Definition and Minimal Pairs Szczecin, 8.05.2018 5 / 17

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Minimal pairs – different approach

Let w be a given extension of v to K (x) and w an extension of w toK (x). Consider the set

w(x − K ) := {w(x − a) | a ∈ K}.

Theorem 1, [3]

w is a residue transcendental extension if and only if:1 v K = w K (x),2 the set w(x − K ) is upper bounded in w K (x),3 w K (x) contains its upper bound.

Let δ be the upper bound of w(x − K ). Then there exists α ∈ K suchthat δ = w(x − α) and thus ([3]) w is a residue transcendental extensionof v defined by (1). The pair (α, δ) is called a pair of definition.

H.Ćmiel,P.Szewczyk (University of Szczecin) Pairs of Definition and Minimal Pairs Szczecin, 8.05.2018 5 / 17

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DefinitionA pair of definition (α, δ) is called minimal (or minimal relative to K ) if itis a minimal pair in the sense of the previous definition.Let w1,w2 be two residue transcendental extensions of v to K (x). We saythat w2 dominates w1 (written w1 ≤ w2) if w1

(f (x)

)≤ w2

(f (x)

)for all

polynomials f ∈ K [x ]. If w2 ≥ w1 and there exists f ∈ K [x ] such thatw1(f ) < w2(f ), we say that w2 well dominates w1, which we will denote asw1 < w2.

Proposition 1, [4]

Let K be algebraically closed and let w1,w2 be two residue transcendentalextensions of v to K (x). Let (αi , δi ) be a pair of definition of wi , i = 1, 2.The following statements are equivalent:1 w1 ≤ w22 δ1 ≤ δ2 and v(α1 − α2) ≥ δ1.

Moreover, w1 < w2 if and only if δ1 < δ2 and v(α1 − α2) ≥ δ1.

H.Ćmiel,P.Szewczyk (University of Szczecin) Pairs of Definition and Minimal Pairs Szczecin, 8.05.2018 6 / 17

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DefinitionA pair of definition (α, δ) is called minimal (or minimal relative to K ) if itis a minimal pair in the sense of the previous definition.Let w1,w2 be two residue transcendental extensions of v to K (x). We saythat w2 dominates w1 (written w1 ≤ w2) if w1

(f (x)

)≤ w2

(f (x)

)for all

polynomials f ∈ K [x ]. If w2 ≥ w1 and there exists f ∈ K [x ] such thatw1(f ) < w2(f ), we say that w2 well dominates w1, which we will denote asw1 < w2.

Proposition 1, [4]

Let K be algebraically closed and let w1,w2 be two residue transcendentalextensions of v to K (x). Let (αi , δi ) be a pair of definition of wi , i = 1, 2.The following statements are equivalent:1 w1 ≤ w22 δ1 ≤ δ2 and v(α1 − α2) ≥ δ1.

Moreover, w1 < w2 if and only if δ1 < δ2 and v(α1 − α2) ≥ δ1.

H.Ćmiel,P.Szewczyk (University of Szczecin) Pairs of Definition and Minimal Pairs Szczecin, 8.05.2018 6 / 17

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By an ordered system of residue transcendental extensions of v to K (x)(for brevity call it an ordered system) we mean a family (wi )i∈I of residuetranscendental extensions of v to K (x), where I is a well ordered setwithout a last element and such that wj dominates wi when i < j .

For an ordered system (wi )i∈I and any given f ∈ K [x ] let us define themapping

w(f ) := supi∈I

wi (f ).

As stated in [4], w is a valuation on K [x ]. It will be called the limit of thegiven system (wi )i∈I and denoted by w = supi wi .For each i ∈ I we denote by (αi , δi ) a pair of definition of wi . Then byProposition 1, the set (δi )i∈I is a well ordered subset of vK . Moreover, iffor every i , j ∈ I , i < j , wj well dominates wi , then (αi )i∈I is apseudo-convergent sequence on K .

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By an ordered system of residue transcendental extensions of v to K (x)(for brevity call it an ordered system) we mean a family (wi )i∈I of residuetranscendental extensions of v to K (x), where I is a well ordered setwithout a last element and such that wj dominates wi when i < j .For an ordered system (wi )i∈I and any given f ∈ K [x ] let us define themapping

w(f ) := supi∈I

wi (f ).

As stated in [4], w is a valuation on K [x ]. It will be called the limit of thegiven system (wi )i∈I and denoted by w = supi wi .

For each i ∈ I we denote by (αi , δi ) a pair of definition of wi . Then byProposition 1, the set (δi )i∈I is a well ordered subset of vK . Moreover, iffor every i , j ∈ I , i < j , wj well dominates wi , then (αi )i∈I is apseudo-convergent sequence on K .

H.Ćmiel,P.Szewczyk (University of Szczecin) Pairs of Definition and Minimal Pairs Szczecin, 8.05.2018 7 / 17

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By an ordered system of residue transcendental extensions of v to K (x)(for brevity call it an ordered system) we mean a family (wi )i∈I of residuetranscendental extensions of v to K (x), where I is a well ordered setwithout a last element and such that wj dominates wi when i < j .For an ordered system (wi )i∈I and any given f ∈ K [x ] let us define themapping

w(f ) := supi∈I

wi (f ).

As stated in [4], w is a valuation on K [x ]. It will be called the limit of thegiven system (wi )i∈I and denoted by w = supi wi .For each i ∈ I we denote by (αi , δi ) a pair of definition of wi . Then byProposition 1, the set (δi )i∈I is a well ordered subset of vK . Moreover, iffor every i , j ∈ I , i < j , wj well dominates wi , then (αi )i∈I is apseudo-convergent sequence on K .

H.Ćmiel,P.Szewczyk (University of Szczecin) Pairs of Definition and Minimal Pairs Szczecin, 8.05.2018 7 / 17

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Theorem 2, [4]

Let K be a field, and let (wi )i∈I be an ordered system of residuetranscendental extensions of v to K (x). For every i ∈ I we denote by(αi , δi ) a fixed minimal pair of definition of wi with respect to K . Denoteby wi the restriction of wi to K (x) and by vi the restriction of v to K (αi ),i ∈ I . Then

a) For all i , j ∈ I , j < j one has wi < wj , i.e. (wi )i∈I is an orderedsystem of residue transcendental extensions of v to K (x).

b) For all i , j ∈ I , i < j one has Kvi ⊆ Kvj and viK ⊆ vjK .

c) Assume that w = sup wi and w is not a residue transcendentalextension of v to K (x). Let w be the restriction of w to K (x). Thenw = supi wi . Moreover, one has

Kw =⋃i∈I

Kvi and wK =⋃i∈I

viK .

H.Ćmiel,P.Szewczyk (University of Szczecin) Pairs of Definition and Minimal Pairs Szczecin, 8.05.2018 8 / 17

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Theorem 3, [4]

Let w be a given value transcendental extension of v to K (x). Consider acofinal well ordered set {δi | i ∈ I} ⊆ vK and some αi such that

w(x − αi ) = δi , i ∈ I .

Let wi = wαiδi . Then

a) wi < wj if i < j , i.e. {wi}i∈I is an ordered system of residuetranscendental extensions of v to K (x). Moreover, for every i < j wj

well dominates wi .

b) wi ≤ w for all i ∈ I and w = supi∈I wi .

H.Ćmiel,P.Szewczyk (University of Szczecin) Pairs of Definition and Minimal Pairs Szczecin, 8.05.2018 9 / 17

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Theorem 4, [4]

Let w be a value transcendental extension of v to K (x). Then there existsa pair (α, δ) ∈ K × wK (x) such that w(x − α) = δ. Moreover,wK (x) = vK ⊕ Zδ and w is defined by

w

( n∑i=0

ai (x − α)i)

= infi

(v(ai ) + iδ

), ai ∈ K . (2)

Conversely, let Γ be an ordered group which contains vK as a subgroup,and δ ∈ Γ be such that Zδ ∩ vK = 0. Let α ∈ K and let w : K (x)→ Γ bedefined by the equality (2). Then w is a value transcendental extension ofv to K (x). Moreover, wK (x) = vK ⊕ Zδ and Kw = Kv .

H.Ćmiel,P.Szewczyk (University of Szczecin) Pairs of Definition and Minimal Pairs Szczecin, 8.05.2018 10 / 17

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Theorem 5, [4]

Let w be a value transcendental extension of v to K (x), let {δi | i ∈ I} bea set cofinal in w K (x). Choose αi ∈ K , i ∈ I , such that (αi , δi ) areminimal pairs. Take wi to be the restriction of wαiδi to K (x) and vi to bethe restriction of v to K (αi ). Then

wi < wj , Kvi ⊆ Kvj and viK ⊆ vjK whenever i < j .

(wi )i∈I is an ordered system of residue transcendental extensions of vto K (x) and w = supi wi . Moreover, we have

K (x)w =⋃i∈I

Kvi , wK (x) =⋃i∈I

viK .

H.Ćmiel,P.Szewczyk (University of Szczecin) Pairs of Definition and Minimal Pairs Szczecin, 8.05.2018 11 / 17

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Theorem 6, [4]

Let w be a value transcendental extension of v to K (x) and (α, δ) aminimal pair of definition of w with respect to K . Denote by f the monicminimal polynomial of α over K and let γ = w(f ). If g ∈ K [x ] is apolynomial with f -expansion of the form

g =n∑

i=0

gi fi , gi ∈ K [x ], deg gi < deg f ,

thenw(g) = inf

(v(gi (α)

)+ iγ

).

Moreover, if v1 is the restriction of v to K (α), then

K (x)w = K (α)v1 and wK (x) = v1K (α)⊕ Zγ.

H.Ćmiel,P.Szewczyk (University of Szczecin) Pairs of Definition and Minimal Pairs Szczecin, 8.05.2018 12 / 17

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Results on minimal pairs

Given an element α ∈ K , are we able to find δ ∈ v K such that (α, δ) is aminimal pair?

Theorem, [5]

Let (K , v) be henselian.

If α ∈ K is separable over K , then there exists an element δ ∈ v Ksuch that (α, δ) is a minimal pair.

If K is complete with respect to v , then there exists an elementδ ∈ v K such that (α, δ) is a minimal pair.

H.Ćmiel,P.Szewczyk (University of Szczecin) Pairs of Definition and Minimal Pairs Szczecin, 8.05.2018 13 / 17

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Results on minimal pairs

Given an element α ∈ K , are we able to find δ ∈ v K such that (α, δ) is aminimal pair?

Theorem, [5]

Let (K , v) be henselian.

If α ∈ K is separable over K , then there exists an element δ ∈ v Ksuch that (α, δ) is a minimal pair.

If K is complete with respect to v , then there exists an elementδ ∈ v K such that (α, δ) is a minimal pair.

H.Ćmiel,P.Szewczyk (University of Szczecin) Pairs of Definition and Minimal Pairs Szczecin, 8.05.2018 13 / 17

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Results on minimal pairs

Given some extension Γ of vK and some extension k of Kv , can weconstruct an extension w of v to K (x) such that wK (x) = Γ andK (x)w = k?

The following results can be found in [4] and [6].

H.Ćmiel,P.Szewczyk (University of Szczecin) Pairs of Definition and Minimal Pairs Szczecin, 8.05.2018 14 / 17

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Results on minimal pairs

Given some extension Γ of vK and some extension k of Kv , can weconstruct an extension w of v to K (x) such that wK (x) = Γ andK (x)w = k? The following results can be found in [4] and [6].

H.Ćmiel,P.Szewczyk (University of Szczecin) Pairs of Definition and Minimal Pairs Szczecin, 8.05.2018 14 / 17

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Results on minimal pairs

Assume first that (Γ : vK ) <∞ and [k : Kv ] <∞.

Then

a) there exists a value transcendental extension w such that

K (x)w = k and wK (x) = Γ⊕ Zλ (3)

for λ in some group extension for any given ordering;

b) there exists a residue transcendental extension w such that

wK (x) = Γ and K (x)w = k(t). (4)

Conversely,

a) if w is a value transcendental extension then 3 holds;

b) if w is a residue transcendental extension then 4 holds. In particular,K (x)w is a rational function field over a finite extension of Kv (RuledResidue Theorem, [7]).

H.Ćmiel,P.Szewczyk (University of Szczecin) Pairs of Definition and Minimal Pairs Szczecin, 8.05.2018 15 / 17

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Results on minimal pairs

Assume first that (Γ : vK ) <∞ and [k : Kv ] <∞. Then

a) there exists a value transcendental extension w such that

K (x)w = k and wK (x) = Γ⊕ Zλ (3)

for λ in some group extension for any given ordering;

b) there exists a residue transcendental extension w such that

wK (x) = Γ and K (x)w = k(t). (4)

Conversely,

a) if w is a value transcendental extension then 3 holds;

b) if w is a residue transcendental extension then 4 holds. In particular,K (x)w is a rational function field over a finite extension of Kv (RuledResidue Theorem, [7]).

H.Ćmiel,P.Szewczyk (University of Szczecin) Pairs of Definition and Minimal Pairs Szczecin, 8.05.2018 15 / 17

Page 28: Pairs of Definition and Minimal Pairs · minimal pair for every positive 2vKe. H.Ćmiel,P.Szewczyk (University of Szczecin) Pairs of Definition and Minimal Pairs Szczecin, 8.05.2018

Results on minimal pairs

Assume first that (Γ : vK ) <∞ and [k : Kv ] <∞. Then

a) there exists a value transcendental extension w such that

K (x)w = k and wK (x) = Γ⊕ Zλ (3)

for λ in some group extension for any given ordering;

b) there exists a residue transcendental extension w such that

wK (x) = Γ and K (x)w = k(t). (4)

Conversely,

a) if w is a value transcendental extension then 3 holds;

b) if w is a residue transcendental extension then 4 holds. In particular,K (x)w is a rational function field over a finite extension of Kv (RuledResidue Theorem, [7]).

H.Ćmiel,P.Szewczyk (University of Szczecin) Pairs of Definition and Minimal Pairs Szczecin, 8.05.2018 15 / 17

Page 29: Pairs of Definition and Minimal Pairs · minimal pair for every positive 2vKe. H.Ćmiel,P.Szewczyk (University of Szczecin) Pairs of Definition and Minimal Pairs Szczecin, 8.05.2018

Results on minimal pairs

Assume first that (Γ : vK ) <∞ and [k : Kv ] <∞. Then

a) there exists a value transcendental extension w such that

K (x)w = k and wK (x) = Γ⊕ Zλ (3)

for λ in some group extension for any given ordering;

b) there exists a residue transcendental extension w such that

wK (x) = Γ and K (x)w = k(t). (4)

Conversely,

a) if w is a value transcendental extension then 3 holds;

b) if w is a residue transcendental extension then 4 holds. In particular,K (x)w is a rational function field over a finite extension of Kv (RuledResidue Theorem, [7]).

H.Ćmiel,P.Szewczyk (University of Szczecin) Pairs of Definition and Minimal Pairs Szczecin, 8.05.2018 15 / 17

Page 30: Pairs of Definition and Minimal Pairs · minimal pair for every positive 2vKe. H.Ćmiel,P.Szewczyk (University of Szczecin) Pairs of Definition and Minimal Pairs Szczecin, 8.05.2018

Results on minimal pairs

Assume first that (Γ : vK ) <∞ and [k : Kv ] <∞. Then

a) there exists a value transcendental extension w such that

K (x)w = k and wK (x) = Γ⊕ Zλ (3)

for λ in some group extension for any given ordering;

b) there exists a residue transcendental extension w such that

wK (x) = Γ and K (x)w = k(t). (4)

Conversely,

a) if w is a value transcendental extension then 3 holds;

b) if w is a residue transcendental extension then 4 holds. In particular,K (x)w is a rational function field over a finite extension of Kv (RuledResidue Theorem, [7]).

H.Ćmiel,P.Szewczyk (University of Szczecin) Pairs of Definition and Minimal Pairs Szczecin, 8.05.2018 15 / 17

Page 31: Pairs of Definition and Minimal Pairs · minimal pair for every positive 2vKe. H.Ćmiel,P.Szewczyk (University of Szczecin) Pairs of Definition and Minimal Pairs Szczecin, 8.05.2018

Results on minimal pairs

Assume now that Γ ⊇ vK and k ⊇ Kv are countably generated and atleast one of them is infinite.

Then there exists an extension w such that

wK (x) = Γ and K (x)w = k. (5)

Conversely, if (5) holds, then both extensions are countably generated.

H.Ćmiel,P.Szewczyk (University of Szczecin) Pairs of Definition and Minimal Pairs Szczecin, 8.05.2018 16 / 17

Page 32: Pairs of Definition and Minimal Pairs · minimal pair for every positive 2vKe. H.Ćmiel,P.Szewczyk (University of Szczecin) Pairs of Definition and Minimal Pairs Szczecin, 8.05.2018

Results on minimal pairs

Assume now that Γ ⊇ vK and k ⊇ Kv are countably generated and atleast one of them is infinite. Then there exists an extension w such that

wK (x) = Γ and K (x)w = k. (5)

Conversely, if (5) holds, then both extensions are countably generated.

H.Ćmiel,P.Szewczyk (University of Szczecin) Pairs of Definition and Minimal Pairs Szczecin, 8.05.2018 16 / 17

Page 33: Pairs of Definition and Minimal Pairs · minimal pair for every positive 2vKe. H.Ćmiel,P.Szewczyk (University of Szczecin) Pairs of Definition and Minimal Pairs Szczecin, 8.05.2018

Results on minimal pairs

Assume now that Γ ⊇ vK and k ⊇ Kv are countably generated and atleast one of them is infinite. Then there exists an extension w such that

wK (x) = Γ and K (x)w = k. (5)

Conversely, if (5) holds, then both extensions are countably generated.

H.Ćmiel,P.Szewczyk (University of Szczecin) Pairs of Definition and Minimal Pairs Szczecin, 8.05.2018 16 / 17

Page 34: Pairs of Definition and Minimal Pairs · minimal pair for every positive 2vKe. H.Ćmiel,P.Szewczyk (University of Szczecin) Pairs of Definition and Minimal Pairs Szczecin, 8.05.2018

Bibliography

1 S.K. Khanduja. On valuations of K (x).

2 S.K. Khanduja, N. Popescu and K.W. Roggenkamp. On minimalpairs and residually transcendental extensions of valuations.

3 V. Alexandru, N. Popescu and A. Zaharescu. A theorem ofcharacterization of residual transcendental extensions of a valuation.

4 V. Alexandru, N. Popescu and A. Zaharescu. All valuations on K (x).

5 V. Alexandru, N. Popescu and A. Zaharescu. Minimal pairs ofdefinition of a residual transcendental extension of a valuation.

6 F.-V. Kuhlmann. Value groups, residue fields and bad places ofrational function fields.

7 J. Ohm. The Ruled Residue Theorem for simple transcendentalextensions of valued fields.

H.Ćmiel,P.Szewczyk (University of Szczecin) Pairs of Definition and Minimal Pairs Szczecin, 8.05.2018 17 / 17