On Lojasiewicz-type inequalities for exponential polynomials
The authors consider Lojasiewicz inequalities for exponential polynomials in one or several real or complex variables. Such inequalities are lower bounds for the functions in terms of the distance from the zero set. Closely connected with this, in the case of functions of a single variable, are questions about the separation of the zeros from one another. The authors discuss the connections of these questions with transcendental number theory and describe the present `state of the art'. They use the methods of Gelfond for algebraic independence to show that if \[ f(z)=\sum_{\gamma \in \Gamma}e^{\gamma z} p_{\gamma}(z), \] where the \(p_{\gamma}(z)\) are polynomials with algebraic coefficients of which only finitely many are non-zero and \(\Gamma\) is the additive subgroup of \({\mathbb{C}}\) generated by 1,w and w 2 where w is a cube irrationality, then for every \(\epsilon >0\) there exist constants \(c_ 1,c_ 2>0\) so that if \(z_ 1,z_ 2\) are distinct zeros of f then \(| z_ 1-z_ 2| >c_ 1 \exp (-c_ 2| z_ 1|^{4+\epsilon})\). The authors show that if \(f_ 1,...,f_ n\) are n such functions and if V is the set of common zeros then for \(\epsilon >0\), \(a_ 1,a_ 2>0\), there exist \(b_ 1,b_ 2>0\) so that for any z with \[ dist(z,V)>a_ 1 \exp (-a_ 2| z|^{4+\epsilon}) \] one has the Lojasiewicz inequality \[ \sum | f_ j(z)| \quad >\quad b_ 1 \exp (-b_ 2| z|^{4+\epsilon}). \] The authors discuss the case of functions of several variables and pose some problems.
scientific article; zbMATH DE number 4040228
scientific article