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Let X be an algebraic variety equipped with a dominant rational map ϕ : X ⤏ X . A new quantity measuring the interaction of (X , ϕ) with trivial dynamical systems is introduced; the stabilised algebraic dimension of (X , ϕ) captures the maximum number of new algebraically independent invariant rational functions on (X × Y , ϕ × ψ) , as ψ : Y ⤏ Y ranges over all dominant rational maps on algebraic varieties. It is shown that this birational invariant agrees with the maximum dim X ′ where (X , ϕ) ⤏ (X ′ , ϕ ′) is a dominant rational equivariant map and ϕ ′ is part of an algebraic group action on X ′ . As a consequence, it is deduced that if some cartesian power of (X , ϕ) admits a nonconstant invariant rational function, then already the second cartesian power does. [ABSTRACT FROM AUTHOR]

We show that any theory of tame henselian valued fields is NIP if and only if the theory of its residue field and the theory of its value group are NIP. Moreover, we show that if (K, v) is a henselian valued field of residue characteristic char (K v) = p such that if p > 0 , depending on the characteristic of K either the degree of imperfection or the index of the pth powers is finite, then (K, v) is NIP iff Kv is NIP and v is roughly separably tame. [ABSTRACT FROM AUTHOR]

We study definable types in the theory of closed ordered differential fields (CODF). We show a condition for a type to be definable, then we prove that definable types are dense in the Stone space of CODF. [ABSTRACT FROM AUTHOR]

The computable dimension of a structure counts the number of computable copies up to computable isomorphism. In this paper, we consider the possible computable dimensions for various classes of computable ordered fields. We show that computable ordered fields with finite transcendence degree are computably stable, and thus have computable dimension 1. We then build computable ordered fields of infinite transcendence degree which have infinite computable dimension, but also such fields which are computably categorical. Finally, we show that 1 is the only possible finite computable dimension for any computable archimedean field. [ABSTRACT FROM AUTHOR]

We generalize and unify various tools from the study of valued fields equipped with an automorphism to obtain a relative quantifier elimination result for such fields. Along the way we point how the techniques we employ relate to a classical result from tropical geometry. The quantifier elimination result we provide is then applied to the transseries field equipped with the automorphism which sends $$f(x)$$ to $$f(x+1)$$ . [ABSTRACT FROM AUTHOR]

Pour tout entier n, on construit des sous-groupes, infiniment définissables de rang de Lascar ω, du groupe additif d'un corps séparablement clos. [ABSTRACT FROM AUTHOR]

We show that a quadratic form defined over the rational function field ℚ( x 1 , ..., x n ) of dimension at least 4.2 n + 1 is isotropic over all fields ℚ p (x 1 , ..., x n ), except for finitely many primes. Partial results concerning the u-invariant of p-adic function fields are also shown. [ABSTRACT FROM AUTHOR]