Rational solutions of polynomial-exponential equations
type: Item
instance of: scholarly article
zbMATH Open document ID: 1293.11060
DOI: 10.1142/S1793042112500820
author: Ayhan Günaydin
published in: International Journal of Number Theory
publication date: 2012-09-06T00:00:00Z
full work available at URL: https://arxiv.org/abs/1110.6405
Mathematics Subject Classification ID: 11D61
Mathematics Subject Classification ID: 11U09
zbMATH DE Number: 6078179
zbMATH Keywords: polynomial-exponential equation
zbMATH Keywords: rational point
zbMATH Keywords: Schanuel conjecture
zbMATH Keywords: Zilber conjecture
MaRDI profile type: Publication
arXiv ID: 1110.6405
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OpenAlex ID: W3104107370
title: Rational solutions of polynomial-exponential equations
review text: The paper studies the rational solutions of a polynomial-exponential equation over \({\mathbb C}\) of the form \(\sum_{i=1, \dots, s} P_i(X) \, \exp (X \cdot \alpha_i) = 0\) where \(X = (X_1, \dots, X_t)\) is a tuple of indeterminates, \(P_1, \dots, P_s\) are polynomials in \({\mathbb C}[X]\), \(\alpha_1, \dots, \alpha_s \in {\mathbb C}^t\), \(X \cdot \alpha\) abbreviates \(\sum_{j = 1, \dots, t} X_j \alpha_{ij}\) and \(\exp\) is the usual complex exponentiation. A solution \(a = (a_1, \dots, a_t) \in {\mathbb C}^t\) of the previous equation is called \textsl{non degenerate} if \(\sum_{i \in I} P_i(a) \, \exp(a \cdot \alpha_i) \neq 0\) for every proper nonempty subset \(I\) of \(\{ 1, \dots, s \}\). Let \(V = \{ q \in {\mathbb Q}^t \, : \, \forall i \, \forall i' =1, \dots, s\, \, \, q \cdot \alpha_i = q \alpha_{i'} \}\). View \(V\) as a vector space over \({\mathbb Q}\) and consider a complement \(V'\) of \(V\) in \({\mathbb Q}^t\). Denote by \(\pi\), \(\pi'\) the natural projections of \({\mathbb Q^t}\) onto \(V\), \(V'\) respectively.NEWLINENEWLINEThe main result of the paper deals with the non degenerate rational solutions \(q\) of an equation as before, and proves the existence of a positive integer \(N\) such that, for every such \(q\), \(\pi' (q) \in ({1 \over N} {\mathbb Z})^t\).NEWLINENEWLINEMotivations come from Zilber's model theoretic approach to the complex exponential field \(({\mathbb C}, \exp)\) introducing in an axiomatic way certain nice exponential fields -- now called \textsl{Zilber fields} -- and conjecturing that \(({\mathbb C}, \exp)\) is one of them, and actually up to isomorphism the only model of the continuum power. Zilber's axioms include Schanuel's Conjecture, and the main author aim is just deepening the relationship between them and in particular answering the following question positively: Assume Schanuel's Conjecture. Let \(p(Z, Y) \in {\mathbb C}[Z,Y]\) be an irreducible polynomial involving both \(Z\) and \(Y\). Then for every fini…
reviewed by: Carlo Toffalori
description: scientific article; zbMATH DE number 6078179