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Let $P$ and $P'$ two polynomials in $k[[\pi]][t]$ for an algebraically closed field $k$ and let $A=k[[\pi]]$. We consider $ X'=Spec (A[t]/(P'))$ and $X=Spec (A[t]/(P)$.

Let $d=val(\Delta(P))$ where $\Delta(P)$ is the discriminant of $P$.

If the reductions of $P$ and $P'$ are equal modulo $\pi^{2d+1}$, do we have that $X$ is isomorhic to $X'$?

More generally if we replace $k[[\pi]]$ by $R[[\pi]]$ for an artinian ring $R$ and we suppose that $P=P' [\Delta(P)^{2}\pi]$ do we have the same result?

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I fixed the question. – prochet Sep 2 '13 at 9:44

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