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Is the Fujiwara bound the most precise bound on maximum absolute value of complex roots of real polynomials ? Or does it exist some improved version for this special case of real polynomials ?

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This wikipedia page discusses some of the improvements of the Fujiwara bound.

If the coefficients of your polynomial are all positive then the Eneström-Kakeya bound is often very good. See, for example, this paper (PDF). There are some generalizations, for instance when the coefficients lie in a given sector symmetric about the positive real axis.

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