Complex Analysis Proof showing that a Polynomial is linear

In summary, we are given a polynomial P(z) such that P(z) is real if and only if z is real. Our goal is to prove that P(z) is linear. Using the hint provided, we set P = u + iv and z = x+iy, and note that v = 0 if and only if y = 0. From this, we can conclude that 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. Additionally, 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
  • #1
QuantumLuck
19
0

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).
 
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  • #2
anyone have any ideas? after thinking some more i realize that b will follow from cauchy reimann equations so long as i have a. but, i fail to see how a, b, and c together define the polynomial P(z) to be linear or how part a is true.
 
  • #3
Hmmm... I'm not sure off the top of my head, but notice P(z) being real when z is real isn't a very enlightening statement, as any real polynomial satisfies this. So the meat of the proof must lie in P(z) being real only when z is real
 
  • #4
It's really not clear to me how to prove your hint a). But here's another approach. First show P(z) is a polynomial in z with real coefficients. Now from your knowledge of real polynomials, there is a real number A such that P(x)=A has either one or zero REAL roots (depending on whether the highest power of x is odd or even. That means if it's degree is greater than one, it must have a complex root r. Hence?
 
  • #5
Thanks a lot Dick I was able to put together what you said with some of my professor's help to solve the problem. he didn't like the books method either. so i had to show that the power of the polynomial couldn't be even or an odd number that is three or greater. hence, it must be one.
 

Related to Complex Analysis Proof showing that a Polynomial is linear

1. How do you prove that a polynomial is linear using complex analysis?

To prove that a polynomial is linear using complex analysis, we use the fundamental theorem of algebra which states that every polynomial of degree n has exactly n complex roots. If a polynomial has only one root, then it is a linear polynomial.

2. Can a polynomial with complex coefficients be linear?

Yes, a polynomial with complex coefficients can be linear. A linear polynomial is defined as a polynomial of degree 1, and it can have any type of coefficients, whether real or complex.

3. What is the role of the complex plane in proving the linearity of a polynomial?

The complex plane plays a crucial role in proving the linearity of a polynomial. We use the complex plane to graph the polynomial and identify the roots of the polynomial. If the polynomial has only one root, then it is linear.

4. Can a polynomial of degree higher than 1 be linear?

No, a polynomial of degree higher than 1 cannot be linear. A linear polynomial is defined as a polynomial of degree 1, and a polynomial of degree higher than 1 will have more than one root, making it non-linear.

5. Are there any disadvantages to using complex analysis to prove the linearity of a polynomial?

One possible disadvantage of using complex analysis to prove the linearity of a polynomial is that it requires a strong understanding of complex numbers and their properties. It may also involve more complex mathematical calculations compared to other methods of proving linearity.

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