11.5. Appendix A: demonstration of the Schur-Cohn criterion
Let us consider the polynomial:
of degree n ≥ I and its reciprocal polynomial:
We write:
so this is a complex polynomial with two variables x and y.
We observe that . This means that for all , and so on, by writing ah,k = Ah,n−k−1, the application:
is a hermitian quadratic form (HQF).13
EXAMPLE 11.2.– if P = α0(x + α), then .
It seems that for P = a0(x + α), the signature14 of H(P; u1) gives the position of the roots of P in relation to the unit circle. This result is not limited to the polynomials of degree 1, but extends to all the polynomials P(z) as we will see.
We assume that P = P1P2 with p1(x) = b0xr +… +br, and P2(x) =
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