| Let P1 and P2 be polynomials of the requested form:
P1(z) = [p1(z)]^2 + z*[q1(z)]^2
P2(z) = [p2(z)]^2 + z*[q2(z)]^2
or, more briefly,
P1 = p1^2 + z q1^2
P2 = p2^2 + z q2^2
Then
P1 P2 = (p1^2 + z q1^2)(p2^2 + z q2^2)
= p1^2 p2^2 + z (q1^2 p2^2 + p1^2 q2^2) + z^2 q1^2 q2^2
= (p1 p2 + z q1 q2)^2 - 2 z p1 p2 q1 q2
+ z (q1^2 p2^2 + p1^2 q2^2)
= (p1 p2 + z q1 q2)^2
+ z (q1^2 p2^2 - 2 p1 p2 q1 q2 + p1^2 q2^2)
= (p1 p2 + z q1 q2)^2 + z (q1 p2 - p1 q2)^2
Therefor, if P1 and P2 are have the desired form, so does the
product P1 P2.
Now, any zero-degree polynomial P(z) = a can be written as
P(z) = a = (sqrt(a))^2 + z * 0^2
because every complex value a has a square root. Likewise, every
first degree polynomial P(z) = a + bz can be written as
P(z) = a + bz = (sqrt(a))^2 + z * (sqrt(b))^2
So every zeroth or first degree polynomial has the desired form.
Now an arbitrary nth degree polynomial with leading coefficient
a and roots z1, ..., zn can be written as
P(z) = a (z - z1) ... (z - zn)
Since this is a product of polynomials of the desired form, it
follows that P(z) is of the desired form.
So every polynomial in z with complex coefficients can be written
in the form (p(z))^2 + z * (q(z))^2.
Dan
P.S. This is not my solution, it was in a USENET article.
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