scholarly journals Multiplicative valued difference fields

2012 ◽  
Vol 77 (2) ◽  
pp. 545-579 ◽  
Author(s):  
Koushik Pal
Keyword(s):  
Quantifier Elimination ◽  
Real Closed Field ◽  
Closed Field ◽  
General Version ◽  
Special Cases ◽  
Difference Fields ◽  
Valued Difference Fields ◽  
Thomas Scanlon

AbstractThe theory of valued difference fields (K, σ, υ,) depends on how the valuation υ interacts with the automorphism σ. Two special cases have already been worked out - the isometric case, where υ(σ(x)) = υ(x) for all x Є K, has been worked out by Luc Belair, Angus Macintyre and Thomas Scanlon; and the contractive case, where υ(σ(x)) > nυ(x) for all x Є K× with υ(x) > 0 and n Є ℕ, has been worked out by Salih Azgin. In this paper we deal with a more general version, the multiplicative case, where υ(σ(x)) = ρ · υ(x), where ρ (> 0) is interpreted as an element of a real-closed field. We give an axiomatization and prove a relative quantifier elimination theorem for this theory.

A note on valuation definable expansions of fields

1998 ◽  
Vol 63 (2) ◽  
pp. 739-743 ◽  
Author(s):  
Deirdre Haskell ◽  
Dugald Macpherson
Keyword(s):  
Valuation Ring ◽  
Quantifier Elimination ◽  
Real Closed Field ◽  
Closed Field ◽  
Real Closed Fields ◽  
Ordered Field ◽  
Valued Field ◽  
Closed Fields ◽  
Binary Predicate ◽  
Algebraically Closed Fields

In this note, we consider models of the theories of valued algebraically closed fields and convexly valued real closed fields, their reducts to the pure field or ordered field language respectively, and expansions of these by predicates which are definable in the valued field. We show that, in terms of definability, there is no structure properly between the pure (ordered) field and the valued field. Our results are analogous to several other definability results for reducts of algebraically closed and real closed fields; see [9], [10], [11] and [12]. Throughout this paper, definable will mean definable with parameters.Theorem A. Let ℱ = (F, +, ×, V) be a valued, algebraically closed field, where V denotes the valuation ring. Let A be a subset ofFndefinable in ℱv. Then either A is definable in ℱ = (F, +, ×) or V is definable in.Theorem B. Let ℛv = (R, <, +, ×, V) be a convexly valued real closed field, where V denotes the valuation ring. Let Abe a subset ofRndefinable in ℛv. Then either A is definable in ℛ = (R, <, +, ×) or V is definable in.The proofs of Theorems A and B are quite similar. Both ℱv and ℛv admit quantifier elimination if we adjoin a definable binary predicate Div (interpreted by Div(x, y) if and only if v(x) ≤ v(y)). This is proved in [14] (extending [13]) in the algebraically closed case, and in [4] in the real closed case. We show by direct combinatorial arguments that if the valuation is not definable then the expanded structure is strongly minimal or o-minimal respectively. Then we call on known results about strongly minimal and o-minimal fields to show that the expansion is not proper.

ON DIFFERENCE FIELDS WITH QUANTIFIER ELIMINATION

2001 ◽  
Vol 33 (6) ◽  
pp. 641-646 ◽  
Author(s):  
RAHIM MOOSA
Keyword(s):  
Quantifier Elimination ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Integer Power ◽  
Frobenius Automorphism ◽  
Difference Field ◽  
Difference Fields

This paper proves that a difference field (E, σ) admits quantifier elimination if and only if E is an algebraically closed field, and σ is an integer power of the Frobenius automorphism.

scholarly journals On the Cardinality of A Semi-Algebraic Set

1994 ◽  
Vol 1 (3) ◽  
pp. 277-286
Author(s):  
G. Khimshiashvili
Keyword(s):  
Quadratic Forms ◽  
Real Closed Field ◽  
Closed Field ◽  
Algebraic Subset ◽  
Algebraic Set

Abstract It is shown that the cardinality of a finite semi-algebraic subset over a real closed field can be computed in terms of signatures of effectively constructed quadratic forms.

Equations and Complexity for the Dubois–Efroymson Dimension Theorem

2009 ◽  
Vol 52 (2) ◽  
pp. 224-236
Author(s):  
Riccardo Ghiloni
Keyword(s):  
Real Closed Field ◽  
Closed Field ◽  
Algebraic Subset ◽  
Algebraically Closed Field ◽  
Algebraic Set ◽  
Real Algebraic Set ◽  
Nonsingular Point ◽  
Minimum Integer ◽  
The Ideal

AbstractLetRbe a real closed field, letX⊂Rnbe an irreducible real algebraic set and letZbe an algebraic subset ofXof codimension ≥ 2. Dubois and Efroymson proved the existence of an irreducible algebraic subset ofXof codimension 1 containingZ. We improve this dimension theorem as follows. Indicate by μ the minimum integer such that the ideal of polynomials inR[x1, … ,xn] vanishing onZcan be generated by polynomials of degree ≤ μ. We prove the following two results: (1) There exists a polynomialP∈R[x1, … ,xn] of degree≤ μ+1 such thatX∩P–1(0) is an irreducible algebraic subset ofXof codimension 1 containingZ. (2) LetFbe a polynomial inR[x1, … ,xn] of degreedvanishing onZ. Suppose there exists a nonsingular pointxofXsuch thatF(x) = 0 and the differential atxof the restriction ofFtoXis nonzero. Then there exists a polynomialG∈R[x1, … ,xn] of degree ≤ max﹛d, μ + 1﹜ such that, for eacht∈ (–1, 1) \ ﹛0﹜, the set ﹛x∈X|F(x) +tG(x) = 0﹜ is an irreducible algebraic subset ofXof codimension 1 containingZ. Result (1) and a slightly different version of result (2) are valid over any algebraically closed field also.

The model theory of ordered differential fields

1978 ◽  
Vol 43 (1) ◽  
pp. 82-91 ◽  
Author(s):  
Michael F. Singer
Keyword(s):  
Meromorphic Functions ◽  
Analytic Solutions ◽  
Positive Element ◽  
Real Closed Field ◽  
Real Field ◽  
Closed Field ◽  
Ordered Field ◽  
Positive Elements ◽  
Odd Degree ◽  
Differential Fields

In this paper, we show that the theory of ordered differential fields has a model completion. We also show that any real differential field, finitely generated over the rational numbers, is isomorphic to some field of real meromorphic functions. In the last section of this paper, we combine these two results and discuss the problem of deciding if a system of differential equations has real analytic solutions. The author wishes to thank G. Stengle for some stimulating and helpful conversations and for drawing our attention to fields of real meromorphic functions.§ 1. Real and ordered fields. A real field is a field in which −1 is not a sum of squares. An ordered field is a field F together with a binary relation < which totally orders F and satisfies the two properties: (1) If 0 < x and 0 < y then 0 < xy. (2) If x < y then, for all z in F, x + z < y + z. An element x of an ordered field is positive if x > 0. One can see that the square of any element is positive and that the sum of positive elements is positive. Since −1 is not positive, an ordered field is a real field. Conversely, given a real field F, it is known that one can define an ordering (not necessarily uniquely) on F [2, p. 274]. An ordered field F is a real closed field if: (1) every positive element is a square, and (2) every polynomial of odd degree with coefficients in F has a root in F. For example, the real numbers form a real closed field. Every ordered field can be embedded in a real closed field. It is also known that, in a real closed field K, polynomials satisfy the intermediate value property, i.e. if f(x) ∈ K[x] and a, b ∈ K, a < b, and f(a)f(b) < 0 then there is a c in K such that f(c) = 0.

scholarly journals Continued Fractions Related to $(t,q)$-Tangents and Variants

10.37236/2014 ◽  
2011 ◽  
Vol 18 (2) ◽  
Author(s):  
Helmut Prodinger
Keyword(s):  
Continued Fractions ◽  
Continued Fraction ◽  
General Version ◽  
Continued Fraction Expansion ◽  
Special Cases ◽  
Tangent Function

For the $q$-tangent function introduced by Foata and Han (this volume) we provide the continued fraction expansion, by creative guessing and a routine verification. Then an even more recent $q$-tangent function due to Cieslinski is also expanded. Lastly, a general version is considered that contains both versions as special cases.

The Pierce-Birkhoff Conjecture for Curves

1992 ◽  
Vol 44 (6) ◽  
pp. 1262-1271 ◽  
Author(s):  
Murray Marshall
Keyword(s):  
Real Closed Field ◽  
Noetherian Rings ◽  
Closed Field ◽  
Polynomial Functions ◽  
Piecewise Polynomial ◽  
Algebraic Set ◽  
Dedekind Domains ◽  
Piecewise Polynomial Functions ◽  
Relative Case

AbstractThe results obtained extend Madden’s result for Dedekind domains to more general types of 1-dimensional Noetherian rings. In particular, these results apply to piecewise polynomial functions t:C → R where R is a real closed field and C ⊆ Rn is a closed 1-dimensional semi-algebraic set, and also to the associated “relative” case where t, C are defined over some subfield K ⊆ R.

Differentially algebraic group chunks

1990 ◽  
Vol 55 (3) ◽  
pp. 1138-1142 ◽  
Author(s):  
Anand Pillay
Keyword(s):  
Algebraic Group ◽  
Characteristic Zero ◽  
Quantifier Elimination ◽  
Closed Field ◽  
Zariski Topology ◽  
First Order ◽  
Differential Field ◽  
Closed Fields ◽  
Differential Fields ◽  
Algebraically Closed Fields

We point out that a group first order definable in a differentially closed field K of characteristic 0 can be definably equipped with the structure of a differentially algebraic group over K. This is a translation into the framework of differentially closed fields of what is known for groups definable in algebraically closed fields (Weil's theorem).I restrict myself here to showing (Theorem 20) how one can find a large “differentially algebraic group chunk” inside a group defined in a differentially closed field. The rest of the translation (Theorem 21) follows routinely, as in [B].What is, perhaps, of interest is that the proof proceeds at a completely general (soft) model theoretic level, once Facts 1–4 below are known.Fact 1. The theory of differentially closed fields of characteristic 0 is complete and has quantifier elimination in the language of differential fields (+, ·,0,1, −1,d).Fact 2. Affine n-space over a differentially closed field is a Noetherian space when equipped with the differential Zariski topology.Fact 3. If K is a differentially closed field, k ⊆ K a differential field, and a and are in k, then a is in the definable closure of k ◡ iff a ∈ ‹› (where k ‹› denotes the differential field generated by k and).Fact 4. The theory of differentially closed fields of characteristic zero is totally transcendental (in particular, stable).

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