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QuantumLuck
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Homework Statement
Suppose P is a polynomial such that P(z) is real iff. z is real. Prove that P is linear.
The hint given in the text is to set P = u + iv, z = x+iy and note that v = 0 iff y = 0.
We are then told to conclude that
a. either v-sub y(partial of v with respect to y) is greater than or equal to 0 throughout the real axis or is less than or equal to 0 throughout the real axis;
b. either u-sub x(partial of u with respect to x) is greater than or equal to zero or is less than or equal to zero for all real values and hence u is monotonic along the real-axis;
c. P(z) = alpha has only one solution for real-valued alpha.
Homework Equations
Cauchy-Reimann equations.
Fundamental Theorem of Algebra (FTA) (i think)
The Attempt at a Solution
My friend was of the opinion that we can say v > 0 for y > 0, then we must have v < 0 for y < 0 because the FTA says v greater than or equal to 0 for all z is impossible. (yet i do not see how this follows at all and i am not sure the path to a solution that this will bring.
I am not quite sure how the proof is supposed to follow from the given guidelines and any help would be much appreciated (even a hint).