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Maths > Algebraic geometry > Hensel's lemma

Hensel's lemma and algebraic functions

Posted by Martin Orr on Monday, 05 April 2010 at 22:34

An algebraic function is a function which we obtain by solving a polynomial in two variables x and y to write y as a function of x. In general polynomials have more than one root, so (informally) we get a multi-valued function. In this post I will restrict attention to regions of the plane in which we can unambiguously pick a single "branch" of the function. What happens where branches meet will be the subject of a later post.

I shall give an algorithm for expressing an algebraic function (in such a nicely behaved region) as a power series, thereby proving that a power series solution to the original polynomial exists. The generalisation of this result to a complete discrete valuation ring is Hensel's lemma, and is particularly important to number theorists in the case of p-adic integers (which were invented by Hensel). In this post I will focus on the case of algebraic functions, as it is easier to apply geometric intuition.

Algorithm for constructing power series

Suppose we have a polynomial f(x, y) \in \mathbb{C}[x, y], which defines a curve in the affine plane. We want to solve the equation f(x, y) = 0 to obtain y as a power series in x. (I am interested in formal solutions here, and don't care about convergence of the power series.)

Of course, in general there will be several values of y corresponding to a single x, so there are several power series which solve the equation.

Consider a value y_0 at which g(y) := f(0, y) has a simple zero (i.e. g'(y_0) \neq 0). Then as we move along the curve near (0, y_0), y is a single-valued function of x, so we can expect to look for a single power series solution with constant term y_0.

Of course it is not obvious that the function y(x) can actually be expressed by a power series. This will follow from giving an algorithm to construct a power series solution (ignoring convergence issues).

Curve y^3 - y^2 - x = 0
y^3 - y^2 - x = 0

For example, take f(x, y) = y^3 - y^2 - x.

Then f(0, y) = y^3 - y^2 has a simple zero at y_0 = 1.

So there should be a unique power series y = 1 + a_1 x + a_2 x^2 + ... satisfying f(x, y) = 0.

To find it, we can solve for a_i one at a time:

  1. Try y_1 = 1 + a_1 x. Substitute this in f(x, y) and we get a polynomial in x with no constant term. The x term is (a_1 - 1) x so we set a_1 = 1 to make this term vanish.

  2. Try y_2 = 1 + x + a_2 x^2. Substituting this in f(x, y), we get a polynomial where the constant and x terms vanish. The x^2 term is (a_2 + 2) x^2 so setting a_2 = -2, this term vanishes.

  3. Try y_3 = 1 + x - 2x^2 + a_3 x^3. Substituting this in f(x, y), we get a polynomial where the constant, x and x^2 terms vanish. The x^3 term is (a_3 - 7) x^3 so setting a_3 = 7, this term vanishes.

Curve y^3 - y^2 - x = 0

Continuing in this way, we build up the power series y_\infty = 1 + x - 2x^2 + 7x^3 - 30x^4 + 143x^5 - 728x^6 + \ldots

The graph to the right shows that the first four terms of this power series approximate the curve f(x, y) = 0 near (0, 1).


Proof that the algorithm works

Now we shall prove that the algorithm does construct a power series y_\infty satisfying f(x, y_\infty) = 0.

At each step we are setting y_k to be a polynomial of degree k, with unknown x^k coefficient and other coefficients known.

By induction, we have chosen a_0, \ldots, a_{k-1} so that the terms of degree less than k in the expansion of f(x, y_{k-1}) vanish. The a_k x^k term in y_k does not affect terms of lower degree in f(x, y_k), so terms of degree less than k in f(x, y_k) vanish.

We need to show that there is a unique value for a_k such that the x^k term of f(x, y_k) vanishes. But this term has the form (g'(y_0) a_k + c) x^k for some constant c, so indeed the condition g'(y_0) \neq 0 ensures that there is a unique a_k making it vanish.

If we let y_\infty be the power series obtained as the limit of the y_k, then the terms of f(x, y_\infty) of degree at most k are the same as the terms of f(x, y_k) of degree at most k, so are all 0. Since this holds for all k, f(x, y_\infty) = 0 as formal power series.

(Note: it is perhaps wrong to call this an "algorithm" since it doesn't terminate. It can't terminate because it is constructing an infinite object, a power series. But in finite time it will give us any particular term of the power series.)

Hensel's lemma

The above algorithm gives a constructive proof of:

Hensel's lemma. Let f(x, y) be a polynomial in \mathbb{C}[x, y], and suppose that y_0 is a simple root of g(y) := f(0, y). Then there is a unique y(x) \in \mathbb{C}[[x]] (the ring of formal power series) such that y(0) = y_0 and f(x, y(x)) = 0 as formal power series.

Note that the lemma is entirely asymmetric between x and y. We can look at it as solving a polynomial in one variable y, whose coefficients happen to be elements of the ring \mathbb{C}[x]. Since the solutions end up in \mathbb{C}[[x]], it is more natural to consider polynomials in y with coefficients in \mathbb{C}[[x]] (the resulting functions y(x) are sometimes called algebroid functions). Exactly the same algorithm still works.

If you know what a complete discrete valuation ring is, you can observe that the algorithm works for polynomials over any complete DVR (the function g(y), generalising f(0, y), being the reduction of f mod the maximal ideal). Any DVR A is a 1-dimensional ring, so A[y] is 2-dimensional. You can look at Hensel's lemma as being about giving a local expression for y on the curve f(y) = 0, near a point (*, y_0) (where * is the closed point of \mathop{\mathrm{Spec}} A). The condition that g'(y_0) \neq 0 is the condition that near this point, the curve has just one branch.

Tags alg-geom, maths, number-theory

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