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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]