Uniqueness of Hensel factors of a polynomial (invariant to change of "basepoint")?

Question

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!

Answer 1

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 $g_i^{(b)}(X-b) = g_i^{(b')}(X-b')$.