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Fields of surreal numbers and exponentiation

Lou van den Dries, Philip Ehrlich (2001)

Fundamenta Mathematicae

We show that Conway's field of surreal numbers with its natural exponential function has the same elementary properties as the exponential field of real numbers. We obtain ordinal bounds on the length of products, reciprocals, exponentials and logarithms of surreal numbers in terms of the lengths of their inputs. It follows that the set of surreal numbers of length less than a given ordinal is a subfield of the field of all surreal numbers if and only if this ordinal is an ε-number. In that case,...

Finite-dimensional differential algebraic groups and the Picard-Vessiot theory

Anand Pillay (2002)

Banach Center Publications

We make some observations relating the theory of finite-dimensional differential algebraic groups (the ∂₀-groups of [2]) to the Galois theory of linear differential equations. Given a differential field (K,∂), we exhibit a surjective functor from (absolutely) split (in the sense of Buium) ∂₀-groups G over K to Picard-Vessiot extensions L of K, such that G is K-split iff L = K. In fact we give a generalization to "K-good" ∂₀-groups. We also point out that the "Katz group" (a certain linear algebraic...

Finiteness results for Hilbert's irreducibility theorem

Peter Müller (2002)

Annales de l’institut Fourier

Let k be a number field, 𝒪 k its ring of integers, and f ( t , X ) ∈ k ( t ) [ X ] be an irreducible polynomial. Hilbert’s irreducibility theorem gives infinitely many integral specializations t ↦ t ¯ ∈ 𝒪 k such that f ( t ¯ , X ) is still irreducible. In this paper we study the set Red f ( 𝒪 k ) of those t ¯ ∈ 𝒪 k with f ( t ¯ , X ) reducible. We show that Red f ( 𝒪 k ) is a finite set under rather weak assumptions. In particular, previous results obtained by diophantine approximation techniques, appear as special cases of some of our results. Our method is different. We use elementary group...

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