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We consider the extremes of the logarithm of the characteristic polynomial ofmatrices from the C$\beta$E ensemble. We prove convergence in distribution ofthe centered maxima (of the real and imaginary parts) towards the sum of aGumbel variable and another independent variable, which we characterize as thetotal mass of a "derivative martingale". We also provide a description of thelandscape near extrema points.

The extremal landscape for the C$\beta$E ensemble