Valuations and Dedekind's Prague Theorem

Coquand, Thierry; Persson, Henrik

doi:10.1016/s0022-4049(99)00095-x

Cited by 29 publications

(27 citation statements)

References 7 publications

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“…[30]). The paper [5] of Coquand and Persson also contains another kind of formal "IntegralvalueStellensatz". The direct part of Theorem 3.1 in [30] has the following consequence.…”

Section: Related Results Of Prestel-ripolimentioning

confidence: 99%

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Dynamical method in algebra: effective Nullstellensätze

Coste

Lombardi

Roy

2001

Annals of Pure and Applied Logic

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We give a general method for producing various effective Null and Positivstellensätze, and getting new Positivstellensätze in algebraically closed valued fields and ordered groups. These various effective Nullstellensätze produce algebraic identities certifying that some geometric conditions cannot be simultaneously satisfied. We produce also constructive versions of abstract classical results of algebra based on Zorn's lemma in several cases where such constructive version did not exist. For example, the fact that a real field can be totally ordered, or the fact that a field can be embedded in an algebraically closed field. Our results are based on the concepts we develop of dynamical proofs and simultaneous collapse. MSC: 03F65, 06F15, 12J10, 12J15, 18B25

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“…[30]). The paper [5] of Coquand and Persson also contains another kind of formal "IntegralvalueStellensatz". The direct part of Theorem 3.1 in [30] has the following consequence.…”

Section: Related Results Of Prestel-ripolimentioning

confidence: 99%

“…The fact T > 0 follows from P (1) and P (2). Under this node there is another node [rd], where the inverse of T has been added according to P (5).…”

Section: Dynamical Theories and Dynamical Proofsmentioning

confidence: 99%

Dynamical method in algebra: effective Nullstellensätze

Coste

Lombardi

Roy

2001

Annals of Pure and Applied Logic

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“…Alternatively one can prove NC int by means of a trick that has been ascribed to Gauß [15,29], and which is nothing but an explicit version of (6.6). To this end let again R be a domain, and suppose that f g = 1.…”

Section: Nilpotent Coefficients With Finite Inductionmentioning

confidence: 99%

Induction in Algebra: A First Case Study

Schuster

2012

2012 27th Annual IEEE Symposium on Logic in Computer Science

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Abstract. Many a concrete theorem of abstract algebra admits a short and elegant proof by contradiction but with Zorn's Lemma (ZL). A few of these theorems have recently turned out to follow in a direct and elementary way from the Principle of Open Induction distinguished by Raoult. The ideal objects characteristic of any invocation of ZL are eliminated, and it is made possible to pass from classical to intuitionistic logic. If the theorem has finite input data, then a finite partial order carries the required instance of induction, which thus is constructively provable. A typical example is the well-known theorem "every nonconstant coefficient of an invertible polynomial is nilpotent".

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“…This is a consequence of Kronecker's theorem [3,5,6] that states that if P 1 P 2 = Q ∈ A[x] then any product u 1 u 2 , where u i is a coefficient of P i , is integral over the ring generated by the coefficients of Q. Since g 1 (0) = 1, this implies that u is integral over A.…”

Section: Remarkmentioning

confidence: 99%