Hadamard Factorization of Stable Polynomials

The stable (Hurwitz) polynomials are important in the study of differential equations systems and control theory (see [7] and [19]). A property of these polynomials is related to Hadamard product. Consider two polynomials p; q 2 is an element of R [x] : p(x) = a(n)x(n) + a(n-1)x(n-1) + ... + a(1)x+a(0) q(x) = b(m)x(m) + b(m-1)x(m-1) + ... + b(1) x + b(0) the Hadamard product (p * q) is defined as (p * q) (x) = a(k)b(k)x(k) + a(k-1)b(k-1)x(k-1) + ... + a(1)b(1)x + a(0)b(0) where k = min (m, n). Some results (see [16]) shows that if p, q is an element of R [x] are stable polynomials then (p * q) is stable, also, i.e. the Hadamard product is closed; however, the reciprocal is not always true, that is, not all stable polynomial has a factorization into two stable polynomials the same degree n, if n >= 4 (see [15]). In this work we will give some conditions to Hadamard factorization existence for stable polynomials.