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A strengthening of Siegel's proof of the Hermite-Lindemann Theorem is given. The results are used to obtain lower bounds for the values of exponential polynomials at algebraic points. ITie question of how well the root of an exponential polynomial can be approximated by algebraic numbers is considered, and a lower bound is obtained for the absolute value of the difference between a root of the exponential polynomial and an algebraic number.doi:10.2307/1996549 fatcat:hiewo3fymjfcxbpmtvwlm35t7u