Rational solutions of polynomial-exponential equations (Q2907088)

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 finite subset \(A\) of \({\mathbb C}\) there is a generic point on the complex curve defined by \(p\) over \({\mathbb Q}(A)\) which is of the form \((\gamma, \exp(\gamma))\). This explains the interest in the polynomial-exponential equations described before.NEWLINENEWLINEComing back to the main theorem, the proof strategy is reducing the original equation to certain classical equations also studied by Laurent. This is done by using other earlier results on multiplicative subgroups of fields. In fact the exponential part of the interesting equation, namely the tuple \((\exp(q \cdot \alpha_i))_i\), varies in a subgroup of \(({\mathbb C}^\times)^s\) of finite rank. The main difficulty in this approach regard the roots of unity. A theorem of Dvornicich and Zannier gives some help in this perspective.NEWLINENEWLINESeveral interesting consequences are also obtained, including a finiteness condition on the solutions of the original equation. It says that, if the \(\exp(\alpha_{ij})\) (\(i = 1, \dots, s\), \(j = 1, \dots, t\)) are multiplicatively independent, then only finitely many non degenerate rational solutions occur.