From Schanuel's conjecture to Shapiro's conjecture
type: Item
instance of: scholarly article
title: From Schanuel's conjecture to Shapiro's conjecture
zbMATH Open document ID: 1378.11103
DOI: 10.4171/CMH/328
author: Paola D'Aquino
author: Giuseppina Terzo
author: Angus Macintyre
published in: Commentarii Mathematici Helvetici
publication date: 2014-10-24T00:00:00Z
full work available at URL: https://arxiv.org/abs/1206.6747
review text: 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 exp…
reviewed by: Luc Bélair
Mathematics Subject Classification ID: 11U09
Mathematics Subject Classification ID: 03C60
Mathematics Subject Classification ID: 30C15
zbMATH DE Number: 6361414
zbMATH Keywords: exponential fields
zbMATH Keywords: Zilber's fields
zbMATH Keywords: Schanuel's conjecture
zbMATH Keywords: Shapiro's conjecture
zbMATH Keywords: anomalous varieties
Wikidata QID: Q123121080
MaRDI profile type: Publication
arXiv ID: 1206.6747
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description: scientific article; zbMATH DE number 6361414