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Prove that for arbitrary polynomials $f,g,h \in k[x,y] (k$ a field),$ (fgh)^2 \in (f^3,g^3,h^3)$(ideal of $k[x,y]$).

This question is an exercise29 from Lectures on Local Cohomology, section 5, by Craig Huneke. And then It said that the local cohomology module $H_{(f,g,h)}^3 (k[x,y]) =0$ by exercise 28 as following:

Let $R$ be a local Noetherian ring of dimension $d$, and let $x_1,...,x_s$ be elements of R generating an ideal $I$. Prove that $H_I^s(R) = 0$ if and only if the following condition is satisfied: for all $p > 0$, there exists $q$ such that $(x_1 ···x_s )^q \in (x_1^{p+q},...,x_s^{p+q}).$

Well
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    Some context, please. How does this question arise? What mathematics do you think might be relevant? Why do you think the statement is true? The more we know, the more likely we are to be able to help. – Gerry Myerson May 09 '23 at 13:24
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    This question is an exercise from Lectures on Local Cohomology, section 5, by Craig Huneke. And then It said that the local cohomology module $H_{(f,g,h)}^3 (k[x,y]) =0$. More information I will add to the discription above. – Well May 09 '23 at 13:35
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    Have you tried doing this when $f,g$ and $h$ are monomials $x^Ay^B$? – Pedro May 09 '23 at 15:24
  • Yes, I have been tried, it is true. – Well May 09 '23 at 15:46

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