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For the sake of simplicity, in the sequel a (Laurent) polynomial of negative degree is understood to be the zero polynomial and, similarly, a banded matrix with negative bandwidth reduces to the zero matrix.
More generally, for a polynomial of an arbitrary degree, the discriminant is zero if and only if it has a multiple root, and, in the case of real coefficients, it is positive if and only if the number of non-real roots is a multiple of 4.