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## Publications

**L'Innocente S, Mantova V** Factorisation of germ-like series *Journal of Logic and Analysis*, **9**, 2017

DOI:10.4115/jla.2017.9.3

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*A classical tool in the study of real closed fields are the fields K((G)) of generalised power series (i.e., formal sums with well-ordered support) with coefficients in a field K of characteristic 0 and exponents in an ordered abelian group G. A fundamental result of Berarducci ensures the existence of irreducible series in the subring K((G≤0)) of K((G)) consisting of the generalised power series with non-positive exponents. It is an open question whether the factorisations of a series in such subring have common refinements, and whether the factorisation becomes unique after taking the quotient by the ideal generated by the non-constant monomials. In this paper, we provide a new class of irreducibles and prove some further cases of uniqueness of the factorisation.*

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**Mantova V** Algebraic equations with lacunary polynomials and the Erdos-Renyi conjecture *Rivista di Matematica della Universita di Parma*, **7**, 239-246, 2016

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*In 1947, Rényi, Kalmár and Rédei discovered some special polynomials p(x)∈C[x] for which the square p(x)2 has fewer non-zero terms than p(x). Rényi and Erdős then conjectured that if the number of terms of p(x) grows to infinity, then the same happens for p(x)2. The conjecture was later proved by Schinzel, strengthened by Zannier, and a 'final' generalisation was proved by C. Fuchs, Zannier and the author. This note is a survey of the known results, with a focus on the applications of the latest generalisation.*

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**Mantova V, Zannier U** Polynomial–exponential equations and Zilber's conjecture *Bulletin of the London Mathematical Society*, **48**, 309-320, 2016

DOI:10.1112/blms/bdv096

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*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.*

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**Mantova V** A pseudoexponential-like structure on the algebraic numbers *The Journal of Symbolic Logic*, **80**, 1339-1347, 2015

DOI:10.1017/jsl.2014.41

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*Pseudoexponential fields are exponential fields similar to complex exponentiation which satisfy the Schanuel Property, i.e., the abstract statement of Schanuel’s Conjecture, and an adapted form of existential closure. Here we show that if we remove the Schanuel Property and just care about existential closure, it is possible to create several existentially closed exponential functions on the algebraic numbers that still have similarities with complex exponentiation. The main difficulties are related to the arithmetic of algebraic numbers, and they can be overcome with known results about specialisations of multiplicatively independent functions on algebraic varieties.*

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**Mantova V, Zannier U** Artin-whaples approximations of bounded degree in algebraic varieties *Proceedings of the American Mathematical Society*, **142**, 2953-2964, 2014

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*© 2014 American Mathematical Society. The celebrated Artin-Whaples approximation theorem (which is a generalization of the Chinese remainder theorem) asserts that, given a field K, distinct places v 1 ,…,v n of K, and points x 1 ,…,x n ∈ℙ1(K), it is possible to find an x ∈ ℙ1(K) simultaneously near x i w.r.t. v i with any prescribed accuracy. If we replace ℙ1 with other algebraic varieties V, the analogous conclusion does not generally hold, e.g., because V may contain too few points over K. However, it has been proved by a number of authors that, at least in the case of global fields, it holds if we allow x to be algebraic over K. These results do not directly contain either the case of ℙ1 or the case of general fields, and above all they do not control the degree of x. In this paper we offer different arguments leading to a general approximation theorem properly generalizing that of Artin-Whaples. This works for every V, K as above, and not only asserts the existence of a suitable (formula presented), but bounds explicitly the degree [K(x) : K] in terms only of geometric invariants of V. It shall also be seen that such a bound is in a sense close to being best-possible.*

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**Mantova V** Approssimazioni di Artin-Whaples con grado limitato su varietà algebriche *Bolletino dell Unione Matematica Italiana*, **6**, 693-697, 2013

**Mantova V** Involutions on Zilber fields *Rendiconti Lincei - Matematica e Applicazioni*, 237-244, 2011

DOI:10.4171/RLM/598

**Berarducci A, Mantova V** Surreal numbers, derivations and transseries *Journal of the European Mathematical Society*

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*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.*

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**Fuchs C, Mantova V, Zannier U** On fewnomials, integral points and a toric version of Bertini's theorem *Journal of the American Mathematical Society*

DOI:10.1090/jams/878

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*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, weachieve 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 extremelyspecial 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 Glm. Such a viewpoint shall lead to some best-possible corollaries in the context of finite covers of Glm, 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.*

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