2013/04/01 by Alberto Lastra, Lastra, Alberto, Stéphane Malek +3
Mathematics · Computer Science · #Meromorphic and Entire Functions #Advanced Differential Equations and Dynamical Systems #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.1304.0334
We investigate the existence of local holomorphic solutions Y of linear partial differential equations in three complex variables whose coefficients are singular along an analytic variety Θ in ℂ2. The coefficients are written as linear combinations of powers of a solution X of some first order nonlinear partial differential equation following an idea we have initiated in a previous work \citemast. The solutions Y are shown to develop singularities along Θ with estimates of exponential type depending on the growth's rate of X near the singular variety. We construct these solutions with the help of series of functions with infinitely many variables which involve derivatives of all orders of X in one variable. Convergence and bounds estimates of these series are studied using a majorant series method which leads to an auxiliary functional equation that contains differential operators in infinitely many variables. Using a fixed point argument, we show that these functional equations actually have solutions in some Banach spaces of formal power series.