Hensel's Lemma - Generalizations

Generalizations

Suppose A is a commutative ring, complete with respect to an ideal, and let be a polynomial with coefficients in A. Then if a ∈ A is an "approximate root" of f in the sense that it satisfies

then there is an exact root b ∈ A of f "close to" a; that is,

and

Further, if f ′(a) is not a zero-divisor then b is unique.

As a special case, if and f ′(a) is a unit in A then there is a unique solution to f(b) = 0 in A such that

This result can be generalized to several variables as follows:

Theorem: Let A be a commutative ring that is complete with respect to an ideal m ⊂ A and f_i(x) ∈ A for i = 1,...,n be a system of n polynomials in n variables over A. Let f = (f₁,...,f_n), viewed as a mapping from An to An, and let J_f(x) be the Jacobian matrix of f. Suppose some a = (a₁, …, a_n) ∈ An is an approximate solution to f = 0 in the sense that

f_i(a) ≡ 0 mod (det J_f(a))2m

for 1 ≤ i ≤ n. Then there is some b = (b₁, …, b_n) in An satisfying f(b) = 0, i.e.,

f_i(b) = 0 for all i,

and furthermore this solution is "close" to a in the sense that

b_i ≡ a_i mod J_f(a)m

for 1 ≤ i ≤ n.

As a special case, if f_i(a) ≡ 0 mod m for all i and det J_f(a) is a unit in A then there is a solution to f(b) = 0 with b_i ≡ a_i mod m for all i.

When n = 1, a = a is an element of A and J_f(a) = J_f(a) is f ′(a). The hypotheses of this multivariable Hensel's lemma reduce to the ones which were stated in the one-variable Hensel's lemma.