# Evaluation of a univariate polynomial based on sum of the coefficients

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 $$g$$.

• Do we have any such polynomial? Yes, for example $P=0$ with $c_i=0$. Aug 23 '20 at 13:42
• Right, but mostly interested in non-trivial cases like non-zero coefficients Aug 23 '20 at 13:51
• Assuming $\sum_i c_i \ne 0$, how about $g(X) = P(X)/\sum_i c_i$? Aug 23 '20 at 13:57
• 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. Aug 23 '20 at 15:03