Stability invariance of discrete and continuous multidimensional systems under some variable tranformations

A variable transformation for stability tests of multidimensional discrete systems, which was previously shown to retain a nonvanishing property of a multivariable polynomial on the distinguished boundary of the multidimensional unit ball, is shown here to retain this property in the entire multidimensional unit ball. Also, a new variable transformation is presented for stability tests of multidimensional continuous systems, and is shown to retain the nonvanishing property on the finite part of the distinguished boundary of the right half hyperplane (the multidimensional finite imaginary axis). Applying this new transformation, some necessary conditions are derived for the nonvanishing of multivariable quadratic form polynomials on the finite multidimensional imaginary axis. Also, explicit necessary and sufficient conditions are derived for this case, where the dimensionality is two.