MathOverflow

I am trying to understand whether there is any existing machinery that could plausibly give cancellation in the following sparse divisor/Kloosterman-type regime.
Let
$$R(h)=h^6+27k^2,$$

I have worked on $m$-symmetric functions, but for the purposes of this post I am mainly interested on these results applied to key polynomials and Demazure atoms. For this reason, I will restrict the discussion to the classical nonsymmetric setting.
First, let me introduce some notation....

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