The Hopf bifurcation theorem gives us a sufficient condition for that there is a Poincaré-Andronov-Hopf bifurcation by using prior assumptions on special coordinates. In 2020, Kruff and Walcher introduced a useful method to compute sufficient conditions for simple Poincaré-Andronov-Hopf bifurcations without such prior assumptions. In the paper, for multiple Hopf bifurcations, we generalize the method. The author has implemented the generalized method on the computer algebra system SageMath. The usefulness of the generalized method is illustrated by the implementation.

Let F= f1(A, X),...,fl(A, X) be a finite set of polynomials in Q[A, X] with variables A=A1,...,Am and X=X1,...,Xn. We study the continuity of the map θ from an element a of Cm to a subset of Cn defined by θ(a)= " the zeros of the polynomial ideal < f1(a, X),..., fl(a, X) >". Let G=(G1, S1),..., (Gk, Sk) be a comprehensive Gröbner system of < F > regarding A as parameters. By a basic property of a comprehensive Gröbner system, when the ideal < f1(a, X),..., fl(a, X) > is zero dimensional for some a ın Si, it is also zero dimensional for any a ın Si and the cardinality of θ(a) is identical on Si counting their multiplicities. In this paper, we prove that θ is also continuous on Si. Our result ensures the correctness of an algorithm for real quantifier elimination one of the authors has recently developed.

A real quantifier elimination method based on the theory of real root counting and the computation of comprehensive Gröbner systems introduced by V. Weispfenning is studied in more detail. We introduce a simpler and more intuitive algorithm which is shown to be an improvement of the original algorithm. Our algorithm is implemented on the computer algebra system Maple using a recent algorithm to compute comprehensive Gröbner systems together with several simplification techniques. According to our computation experiments, our program is superior to other existing implementations for many examples which contain many equalities.