An important component of algorithms for factoring multivariate polynomials over a commutative ring $R$ is *Hensel lifting*. Here's a brief, concrete example to set the stage for my question:


Let $f \in F[X,Y]$, where $F$ is an algebraically closed field. Suppose that for some $b\in F$, $f(b,Y)$ is square-free; call such points *base points*. Let $\alpha^{(b)}\_1,\ldots,\alpha^{(b)}\_n$ be roots of the univariate polynomial $f(b,Y)$, or equivalently roots of $f(X,Y)$ modulo the ideal $(X - b)$. 


Hensel lifting gives an algorithmic way to *lift* the roots $\alpha^{(b)}\_i$ of $f$ modulo $(X - b)$ to roots $g^{(b)}\_i$ modulo $(X-b)^t$ for any $t$, where we have the property that $g^{(b)}\_i = \alpha^{(b)}\_i \mod (X-b)$. Call these roots *Hensel roots*.


(In a polynomial factorization application, there would be a way to take the $g^{(b)}\_i$'s and convert them to actual factors of $f$).




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Here's my question: given two base points $b$ and $b'$ of $f$, we've performed the Hensel lifting procedure to obtain the Hensel factors $\{ g^{(b)}\_i \}$ and $\{ g^{(b')}\_i \}$ (lifted to the same level). Since $F$ is algebraically closed, there are the same numbers of Hensel factors over $b$ as there are over $b'$. 


Is there any relation between $g^{(b)}\_i$ and $g^{(b')}\_i$ that's meaningful? Can we say that $g^{(b)}\_i(X - b) = g^{(b')}\_i(X - b')$? That is, are Hensel roots preserved across change of base points? 


Thank you!



Uniqueness of Hensel factors of a polynomial (invariant to change of "basepoint")? 
Hmm, let's try an example: $f(X,Y) = X - 2 Y + Y^2$ over $\mathbb C$. For any $b \ne 1$ the roots of $f(b,Y)$ are $\alpha\_i^{(b)} = 1 \pm \sqrt{1-b}$. These lift one level to 
$g\_i^{(b)} = -1 \pm \sqrt{1-b} \mp \dfrac{X-b}{2 \sqrt{1-b}}$. They certainly depend on $b$, so it's not true
that $g\_i^{(b)}(X-b) = g\_i^{(b')}(X-b')$. 


