Abstract. We describe the implementation and the use of the package MPSolve (Multiprecision Polynomial Solver) for the approximation of the roots of a univariate polynomial $p(x)=\sum_{i=0}^n a_i x^i$. The algorithm, based on simultaneous approximation of the roots, generates a sequence ${\cal A}_1 \supset {\cal A}_2 \supset \ldots \supset{\cal A}_k$ of nested sets containing the root neighborhood of $p(x)$ defined by the given input precision and outputs Newton-isolated approximations of the roots. The algorithm, particularly suited to deal with sparse polynomials or polynomials defined by straight line programs, adaptively adjusts the working precision to the conditioning of each root so that only the amount of digits of the input coefficients sufficient to recover the requested information on the roots are actually involved in the computations. This feature makes our algorithm particularly suited to deal with polynomials arising from certain symbolic computations where the coefficients are typically integers with hundreds or thousands digits. The present release of the software may perform different computations such as counting, Newton isolation, approximation to any number of digits of the roots belonging to a given subset of the complex plane. Automatic detection of multiplicities and/or of real/imaginary roots can be also performed. Polynomial with approximately known coefficients are also allowed.

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