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We study the existence of formal power series solutions to q-algebraicequations. When a solution exists, we give a sufficient condition on theequation for this solution to have a positive radius of convergence. Weemphasize on the case where the solution is divergent, giving a sharp estimateon the growth of the coefficients. As a consequence, we obtain a bound on theq-Gevrey order of the formal solution, which is optimal in some cases. Variousexamples illustrate our main results.

On q-algebraic equations and their power series solutions