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Math Help - A Novel Bound for Polynomials?

  1. #1
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    A Novel Bound for Polynomials?

    True or false:

    Let f, g, and h \in\mathbb{C}[x] be non-constant, relatively prime polynomials such that f + g = h. Then

    \max (\deg f, \deg g, \deg h) \leq n_0(fgh) - 1,

    where n_0(f) denotes the number of distinct roots of f.
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  2. #2
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    Ok, I have not found an elegant way to approach this but with some intuitive guesses I have found a counterexample showing the claim to be false. Now clearly, from being relatively prime and nonconstant, n_0(fgh)\geq 3, so the first hope for a counter example would be \max (\deg f, \deg g, \deg h)=3, n_0(fgh) =3, in which case the inequality fails. For this to work, each polynomial must be either a perfect square or perfect cube of a linear term. It is not difficult to show that we cannot sum two perfect cubes to another perfect cube by considering the resulting system of equations of the coefficients.

    My solution was constructed by assuming f(x)=(x-a)^3, g(x)=(b-x)^3 and f(x)+g(x)=h(x)=w(x-c)^2, and considering the systems of equations from the coefficients. One set of solutions is a=1, b=\sqrt{3}+2, w=3(\sqrt{3}+3), c=-\sqrt{3}-1, giving the counterexample:

    f(x)=(x+1)^3=x^3+3x^2+3x+1
    g(x)=(\sqrt{3}+2-x)^3=-x^3+3(\sqrt{3}+2)x^2-3(4\sqrt{3}+7))x+(15\sqrt{3}+26)

    h(x)=f(x)+g(x)=3(\sqrt{3}+3)x^2-3(4\sqrt{3}+6))x+(15\sqrt{3}+27)= 3(\sqrt{3}+3)*(x^2-(\sqrt{3}+1)x+(\sqrt{3}+2))=3(\sqrt{3}+3)(x-(\sqrt{3}+1)/2)^2

    So these polynomials, being nonconstant and sharing no roots, satisfy the requirements. However, (fgh)(x)=3(\sqrt{3}+3)(x-(\sqrt{3}+1)/2)^2(x+1)^3(\sqrt{3}+2-x)^3 has three distinct roots, thus 3=\max (\deg f, \deg g, \deg h)=\leq n_0(fgh)-1=3-1=2 which is absurd.
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  3. #3
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    Without giving away the answer to the OP, I will say that I am afraid your argument does not work, siclar, though it was a noble effort!

    In general, we have n_0(fg)\leq n_0(f) + n_0(g), where the equality holds if f and g are relatively prime. So I think your initial step was the problem...
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  4. #4
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    Quote Originally Posted by AlephZero View Post
    True or false:

    Let f, g, and h \in\mathbb{C}[x] be non-constant, relatively prime polynomials such that f + g = h. Then

    \max (\deg f, \deg g, \deg h) \leq n_0(fgh) - 1,

    where n_0(f) denotes the number of distinct roots of f.
    see Mason-Stothers theorem.
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  5. #5
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    Quote Originally Posted by AlephZero View Post
    Without giving away the answer to the OP, I will say that I am afraid your argument does not work, siclar, though it was a noble effort!

    In general, we have n_0(fg)\leq n_0(f) + n_0(g), where the equality holds if f and g are relatively prime. So I think your initial step was the problem...
    Hm, I'm afraid I don't see how this changes anything I wrote, seeing as I explicitly calculated n_0, or rather constructed my polynomials to give it a specific value.

    Having googled NCA's suggestion, I see I must be wrong but I can't figure out where my logic falls apart. I double checked my arithmetic on the polynomial computations and that all seems fine
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