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This might be an easy question but i am sorry for asking this. Let $f(x)\in\mathbb{Z}_p[x].$ Is it always true that $$f(x+y)=f(x)+f'(x)y+f''(x)\frac{y^2}{2}+zy^3$$ for some $z\in\mathbb{Z}_p.$ if it is already given that $y\in\mathbb{Z}_p.$

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  • $\begingroup$ You seem to ask whether it holds that for every polynomial $f$, the rational function in 2 variables $y^{-3}(f(x+y)-f(x)-f'(x)y-f''(x)y^2/2)$ is constant. Why don't you just test monomials $f(x)=x^n$? $\endgroup$
    – YCor
    Commented Mar 5, 2022 at 11:24
  • $\begingroup$ z is not constant here. It should depend on the polynomial and x. $\endgroup$
    – user126352
    Commented Mar 5, 2022 at 11:31
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    $\begingroup$ It's clear that $f$ is fixed. But is $x$ fixed too? The question needs some rephrasing with quantifiers. $\endgroup$
    – YCor
    Commented Mar 5, 2022 at 11:33
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    $\begingroup$ My guess what the question means: Substituting $x+y$ into $f$ and expanding in $y$ gives a formula for $z \in \mathbb{Z}_p[x,y]$. So $z\in\mathbb{Z}_p$ for every $x,y\in\mathbb{Z}_p$. If so the question does not look like suitable for this forum. $\endgroup$
    – Chris Wuthrich
    Commented Mar 5, 2022 at 12:53
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    $\begingroup$ See (2.1) and (2.2) in kconrad.math.uconn.edu/blurbs/gradnumthy/hensel.pdf for the analogue with a last term of degree $2$. To extend that to the third power, as in your question, is straightforward. $\endgroup$
    – KConrad
    Commented Mar 5, 2022 at 19:07

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