[CA] Why Perturbation on Continuous Coefficients of a Complex Polynomial with Distinct Roots has no Influence

cf. Greene Ch5 Problem #18 Question is shown by the title. A nice proof can be done by using Argument principle. We consider the integral form which equals to the number of zeros (subtracted by the number of poles, if the complex function is meromorphic instead of holomorphic), noted by N. This follows the statement of Argument principle. The integrated function is f'/f, where we take f as the polynomial with fixed parameter t0 (for its continuous coefficients). Now perturb t0 to t by a sufficiently small range such that the integrated function f'/f changes a little as well as its integral. However, form N should always be an integer because of its geometric meaning given by Argument principle. That means the little changes caused by parametric perturbation can only be 0 (otherwise 1 or more is too big to be controlled). Hence, if original polynomial under t0 has n distinct roots, then polynomial with parameter t which sufficiently close to t0 must also has n distince roots. The situation does not been affected.