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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).