Abstract. Several authors have conjectured that Conway's field of surreal numbers, equipped with the exponential function of Kruskal and Gonshor, can be described as a field of transseries and admits a compatible differential structure of Hardy-type. In this paper we give a complete positive solution to both problems. We also show that with this new differential structure, the surreal numbers are Liouville closed, namely the derivation is surjective.

We show that Écalle's transseries and their variants (LE and ELseries) can be interpreted as functions from positive infinite surreal numbers to surreal numbers. The same holds for a much larger class of formal series, here called omega-series. Omega-series are the smallest subfield of the surreal numbers containing the reals, the ordinal omega, and closed under the exp and log functions and all possible infinite sums. They form a proper class, can be composed and differentiated, and are surreal analytic. The surreal numbers themselves can be interpreted as a large field of transseries containing the omega-series, but, unlike omega-series, they lack a composition operator compatible with the derivation introduced by the authors in an earlier paper.

Abstract. An old conjecture of Erdős and Rényi, proved by Schinzel, predicted a bound for the number of terms of a polynomial g(x) ∈ C[x] when its square g(x) 2 has a given number of terms. Further conjectures and results arose, but some fundamental questions remained open.In this paper, with methods which appear to be new, we achieve a final result in this direction for completely general algebraic equations f (x, g(x)) = 0, where f (x, y) is monic of arbitrary degree in y, and has boundedly many terms in x: we prove that the number of terms of such a g(x) is necessarily bounded. This includes the previous results as extremely special cases.We shall interpret polynomials with boundedly many terms as the restrictions to 1-parameter subgroups or cosets of regular functions of bounded degree on a given torus G l m . Such a viewpoint shall lead to some best-possible corollaries in the context of finite covers of G l m , concerning the structure of their integral points over function fields (in the spirit of conjectures of Vojta) and a Bertini-type irreducibility theorem above algebraic multiplicative cosets. A further natural reading occurs in non-standard arithmetic, where our result translates into an algebraic and integral-closedness statement inside the ring of non-standard polynomials.

Abstract. Assuming Schanuel's conjecture, we prove that any polynomial exponential equation in one variable must have a solution that is transcendental over a given finitely generated field. With the help of some recent results in Diophantine geometry, we obtain the result by proving (unconditionally) that certain polynomial exponential equations have only finitely many rational solutions.This answers affirmatively a question of David Marker, who asked, and proved in the case of algebraic coefficients, whether at least the one-variable case of Zilber's strong exponential-algebraic closedness conjecture can be reduced to Schanuel's conjecture.