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Do we have any univariate polynomial $P(X)$ with the special property that $P(s)$ for some field element $s$ can be computed based on the sum of the coefficients $c_i$'s and a function of input, say $g(s)$? for example, $P(s) = (\sum_{i=1}^n c_i) \cdot g(s)$ for some arbitrary function $g$.

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  • $\begingroup$ Do we have any such polynomial? Yes, for example $P=0$ with $c_i=0$. $\endgroup$ Aug 23 '20 at 13:42
  • $\begingroup$ Right, but mostly interested in non-trivial cases like non-zero coefficients $\endgroup$
    – Hamidreza
    Aug 23 '20 at 13:51
  • $\begingroup$ Assuming $\sum_i c_i \ne 0$, how about $g(X) = P(X)/\sum_i c_i$? $\endgroup$ Aug 23 '20 at 13:57
  • $\begingroup$ yes, that's true. but I don't want the computation $P(X)$ be included in g(X) again. ...And sorry, I should have composed my question in more detail I think. $\endgroup$
    – Hamidreza
    Aug 23 '20 at 15:03

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