On Isolating Roots in a Multiple Field Extension
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We address univariate root isolation when the polynomial's coefficients are in a multiple field extension. We consider a polynomial F ∈ L[Y], where L is a multiple algebraic extension of ℚ. We provide aggregate bounds for F and algorithmic and bit-complexity results for the problem of isolating its roots. For the latter problem we follow a common approach based on univariate root isolation algorithms. For the particular case where F does not have multiple roots, we achieve a bit-complexity in 𝒪̃_B(n d^2n+2(d+nτ)), where d is the total degree and τ is the bitsize of the involved polynomials.In the general case we need to enhance our algorithm with a preprocessing step that determines the number of distinct roots of F. We follow a numerical, yet certified, approach that has bit-complexity 𝒪̃_B(n^2d^3n+3τ + n^3 d^2n+4τ).
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