For a square matrix A, let S(A) be an eigenvalue inclusion set such as the Gershgorin region, the union of Cassini ovals, and the Ostrowski's set. Characterization is obtained for maps Φ on n × n matrices satisfying S(Φ(A)Φ(B)) = S(AB) for all matrices A and B.

In this paper, we study the quality of positive root bounds. A positive root bound of a polynomial is an upper bound on the largest positive root. Higher quality means that the relative overestimation (the ratio of the bound and the largest positive root) is smaller. We report three findings. (1) Most known positive root bounds can be arbitrarily bad ; that is, the relative over-estimation can approach infinity, even when the degree and the coefficient size are fixed. (2) When the number of sign variations is the same as the number of positive roots, the relative over-estimation of a positive root bound due to Hong (B H) is at most linear in the degree, no matter what the coefficient size is. (3) When the number of sign variations is one, the relative over-estimation of B H is at most constant, in particular 4, no matter what the degree and the coefficient size are.

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Abstract. Polynomial systems of equations are a central object of study in computer algebra. Among the many existing algorithms for solving polynomial systems, perhaps the most successful numerical ones are the homotopy algorithms. The number of operations that these algorithms perform depends on the condition number of the roots of the polynomial system. Roughly speaking the condition number expresses the sensitivity of the roots with respect to small perturbation of the input coefficients. A natural question to ask is how can we bound, in the worst case, the condition number when the input polynomials have integer coefficients? We address this problem and we provide effective bounds that depend on the number of variables, the degree and the maximum coefficient bitsize of the input polynomials. Such bounds allows to estimate the bit complexity of the algorithms that depend on the separation bound, like the homotopy algorithms, for solving polynomial systems.