Extensions and Improvements for the Liu-Xiao's Lemma

The present paper is motivated by a recent result which is obtained by Liu and Xiao, the exact distribution of roots for a quadratic polynomial will be considered. For completeness, the proof of the Liu-Xiao's lemma will firstly be supplemented, however, the condition of the unit complex roots will be improved. In this case, the computations of the Neimark-Sacker bifurcation or the Hopf bifurcation can be simplified. Furthermore, some new results are also established by using the relationship of the roots and coefficients which is very important in the annals of history, through the discoveries of Abel, Galois and Gauss in the nineteenth century. The obtained results can be used for the bifurcations of the two dimensional discrete-time dynamical systems.