fields - An unfamiliar (to me) form of Hensel's Lemma

A far more general result is the "non-archimedean inverse function theorem". I haven't looked at Roquette's reference, so maybe he is mentioning it. But it is something which I didn't really find in the standard number theory textbooks - probably you can find it in texts on $p$-adic analysis - and I learned it from my number theory professor last semester (Jean-Benoît Bost). This theorem is powerful - and I find it fascinating and surprising - and all versions of Hensel's lemma which one usually encounters while learning number theory are immediate consequences.

Let $K$ be a field, $left| cdot right|$ a non-archimedean absolute value on $K$ for which $K$ is complete, $mathcal{O}$ the associated valuation ring, $mathcal{M}$ the maximal ideal, $pi$ a uniformizer. Let $Phi_i in mathcal{O}[X_1,,cdots,X_n]$ for $1 leq i leq n$ and consider the map $Phi = (Phi_1,,cdots,Phi_n) : mathcal{O}^n to mathcal{O}^n$. Let $J$ be the Jacobian $det(partial Phi_i / partial X_j) in mathcal{O}[X_1,,cdots,X_n]$.

Theorem. If $x_0 in mathcal{O}^n$, $y_0 = Phi(x_0)$ and $J(x_0) neq 0$, then for any $R in (0, left|J(x_0)right|)$, $Phi$ induces a bijection

$$overline{B}(x_0,R) to y_0 + (DPhi)(x_0) overline{B}(0,R)$$ (where $DPhi$ is the derivative we all know!) and furthermore we have a bijection $$B^circ(x_0,left|J(x_0)right| to y_0 + (DPhi)(x_0) B^circ(0,left|J(x_0)right|).$$

(I use the standard notations $overline{B}$ and $B^circ$ for closed and open balls respectively.)

The proof uses in an essential way the Picard fixed point theorem.

Corollary 1. Take $n = 1$, $Phi_1 = P$, $x_0 = alpha$, $varepsilon in (0,1)$. Suppose that $left|P(alpha)right| leq varepsilon left|P'(alpha)right|^2$. Then there exists a unique $beta in mathcal{O}$ such that $P(beta) = 0$ and $left|beta - alpharight| leq varepsilon left|P'(alpha)right|$. (We take $R = varepsilon left|P'(alpha)right|$ in the first bijection.)

Hence, as a special case, if $left|P(alpha)right| < left|P'(alpha)right|^2$, we find $left|beta - alpharight| < left|P'(alpha)right|$.

As an even more special case, if $P'(alpha) in mathcal{O}^times$ and $left|P'(alpha)right| <1$, there exists $beta in mathcal{O}$ such that $P(beta) = 0$ and $left|beta - alpharight| < 1$. Restating this in terms of the residue field: a simple zero in the residue field can be lifted to a real zero in $mathcal{O}$. This is the really known version of Hensel's lemma, I guess.

[Definition: the Gauss norm of a polynomial with coefficients in $K$ is defined as the maximum of the absolute values of its coefficients. It is very easy to check that the Gauss norm is multiplicative.]

Corollary 2. Take $f,g,h in mathcal{O}[X]$ such that $deg g = n$, $deg h = m$ and $deg f = deg g + deg h = n + m$. Assume that there exists $varepsilon in (0,1)$ such that $left|f - ghright| leq varepsilonleft|text{Res}(g,h)right|^2$ and $deg(f - gh) leq m + n - 1$. Then there exist $G, H in mathcal{O}[X]$ such that $f = GH$, $deg(G - g) leq n - 1$, $deg(H - h) leq m - 1$, and also $left|G - gright| leq varepsilon left|text{Res}(g,h)right|$ and $left|H - hright| leq varepsilon left|text{Res}(g,h)right|$. (Obviously $text{Res}$ denotes the resultant here, and $left|cdotright|$ the Gauss norm.)

To prove this: write $G = g + xi$ and $H = h + eta$ where $xi$ and $eta$ are polynomials with coefficients in $mathcal{O}$ and have degrees $leq n - 1$ and $leq m - 1$ respectively. Then $f = GH$ if and only if $f = (g + xi)(h + eta)$. It can be seen as a map from $mathcal{O}^n times mathcal{O}^m to mathcal{O}^{n + m}$ given by polynomials. So consider the map $Phi: (xi, eta) mapsto (g + xi)(h + eta) - f$. We have also $text{Res}(g,h) = det((xi, eta) mapsto g xi + h eta))$. It is easy to see that the theorem above then gives the result.

As a corollary: if $f$, $g$ and $h$ satisfy $overline{f} = overline{g} overline{h}$ - where $overline{f}$ is $f$ reduced modulo $mathcal{M}$ et cetera - and if $overline{g}$ and $overline{h}$ are coprime (this is a condition on the resultant!) then there exist $G,H in O[X]$ satisfying the following conditions: $f = GH$, $deg(G - g) leq n - 1$, $deg(H - h)leq m - 1$, $overline{G} = g$ and $overline{H} = h$. Hence "a factorization over the residue field lifts to a factorization over $mathcal{O}$" (under the right conditions).

Corollary 3. Finally, let us come to the motivation for the question: the more general result is that if $P in K[X]$ is irreducible, then $left|Pright|$ (Gauss) is the maximum of the absolute values of the leading coefficient and the constant coefficient. (As a special case, we find the result which Pete L. Clark cites as the Hensel-Kurschak lemma.)

Indeed, let $P(X) = sum_{i = 0}^n a_i X^{n - i} in K[X]$. Suppose WLOG that $left|Pright| = 1$. Let $mathbb{F}$ be the residue field and let $overline{P}$ be the image of $P$ modulo $mathcal{M}$. Set $r = min {n : overline{a_{n - r}} neq 0}$. Then we have in the residue field the factorization $overline{P}(X) = X^r left(overline{a_{n - r}} + overline{a_{n - r - 1}}X + cdots + overline{a_0} X^{n - r}right)$ and we can lift the factorization by Corollary 2, contradicting irreducibility.

I know this is quite some digression; but I find the whole discussion about the various forms of Hensel's lemma very interesting, and I thought this could add something to the discussion.

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