From Schanuel's conjecture to Shapiro's conjecture
Zilber's conjecture on the complex field with exponentiation [\textit{B. Zilber}, Ann. Pure Appl. Logic 132, No. 1, 67--95 (2005; Zbl 1076.03024)] has triggered a renewed interest of model theorists in this fundamental field. Schanuel's conjecture in transcendence theory is a key `` axiom '' in Zilber's approach. The paper under review can be seen as emerging from this circle of ideas. In this paper the authors prove Shapiro's conjecture assuming Schanuel's conjecture; the particular case of this implication for exponential polynomials over the algebraic numbers was proved by \textit{A. C. Shkop} [Commun. Algebra 39, No. 10, 3813--3823 (2011; Zbl 1257.03061)]. For the sake of this review, exponential polynomial functions over \(\mathbb C\) will be of the form \(\lambda_1 e^{\mu_1z}+\ldots + \lambda_n e^{\mu_nz}\), where the \(\lambda_i, \mu_i\) are in \(\mathbb C\). These form a ring \(\mathcal E\) in the natural way. Shapiro's conjecture states that if \(f,g\) are in \(\mathcal E\) and they have infinitely common zeros in \(\mathbb C\), then there exists \(h\) in \(\mathcal E\) such that \(h\) is a common divisor of \(f\) and \(g\) in \(\mathcal E\), and \(h\) has infinitely many zeros in \(\mathbb C\). Recall that Schanuel's conjecture states that if \(\lambda_1, \ldots, \lambda_n\) are in \(\mathbb C\), then the transcendence degree of the field \(\mathbb Q(\lambda_1, \ldots, \lambda_n, e^{\lambda_1}, \ldots, e^{\lambda_n})\) over \(\mathbb Q\) is greater or equal to the linear dimension of \(\lambda_1, \ldots, \lambda_n\) over \(\mathbb Q\). Let's say that an exponential polynomial \(f(z)\) over \(\mathbb C\) is \textit{simple} if \(f(z)=p(e^{\mu z})\) for some standard polynomial \(p\in\mathbb C[X]\) and some \(\mu\in \mathbb C\). The authors work in the general setting of characterisitic zero exponential fields which are algebraically closed fields, and their proof is purely algebraic. The proof naturally splits into two main cases : 1) at least one of the exponential polynomials \(f, g\) is simple; 2) both \(f,g\) are irreducible. Case (1) was known for \(\mathbb C\) \textit{A. J. van der Poorten} and \textit{R. Tijdeman} [Enseign. Math. (2) 21, 57--67 (1975; Zbl 0308.30006)], but the authors provide an algebraic proof bypassing some analytical properties of \(\mathbb C\). Only case (2) requires Schanuel's conjecture, and the proof builds on the work of \textit{E. Bombieri} et al. [Int. Math. Res. Not. 2007, No. 19, Article ID rnm057, 33 p. (2007; Zbl 1145.11049)] and \textit{J. H. Evertse} et al. [Ann. Math. (2) 155, No. 3, 807--836 (2002; Zbl 1026.11038)]. The paper is clearly written.
scientific article; zbMATH DE number 6361414
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