Schwarz mean value theorem and lattice points on analytic curves (after Bombieri and Pila)

Let {{\Gamma\subset {\mathbb R}^2}} be a compact segment of a plane real analytic transcendental curve. Bombieri and Pila (1989) proved that {\Gamma} may not contain “many” rational points with a given denominator. Precisely: given {{\epsilon>0}}, there exists a constant {{c=c(\Gamma, \epsilon)}} such that for any positive integer {N}

\displaystyle  \left|\Gamma \cap (N^{-1}{\mathbb Z})^2\right| \le cN^\epsilon.

(As they remark, the same statement holds if {\Gamma} is algebraic but does not admit polynomial parametrization: this is a rather easy consequence of the Mordell-Weil theorem.)

This theorem is instrumental in the recent novel proof, by Pila and Zannier (2008), of the Manin-Muford conjecture on torsion points on subvarieties of Abelian varieties.

The argument of Bombieri and Pila is based on the mean value theorem of H.A.Schwarz. It asserts that, given an interval {I}, a function {{f\in C^n(I)}}, and {{n+1}} points {{x_0, \ldots, x_n\in I}}, there exists a point {{\tau\in I}} such that

\displaystyle  \det \begin{bmatrix} 1&x_0&\ldots&x_0^{n-1}&f(x_0)\\ &&\ldots\\ 1&x_n&\ldots&x_n^{n-1}&f(x_n) \end{bmatrix}= \frac{f^{(n)}(\tau)}{n!}V(x_0, \ldots, x_n),

where {V(x_0, \ldots, x_n)} is the Vandermond determinant. The case {{n=1}} is the Lagrange mean value theorem.

Dörge (1927) used this theorem to bound the number of integral points on finite segments of algebraic curves and deduce from this a very simple proof of Hilbert’s irreducibility theorem. The work of Dörge is reproduced in Lang’s Fundamentals of Diophantine Geometry (see Theorem 2.1 in Chapter 9), and I advice to read this piece (it is independent of the rest of the book). The argument of Bombieri and Pila is a very beautiful extension of Dörge’s idea.

The starting point of Bombieri and Pila’s argument is yet another “mean value theorem”. Let {I} be an interval, {{x_1, \ldots, x_n\in I}} and {{f_1, \ldots, f_n\in C^{n-1}(I)}}. Then

\displaystyle  \det \bigl[f_i(x_j)\bigr]= \det \left[\frac{f_i^{(j-1)}(\tau_{ij})}{j!}\right]V(x_1, \ldots, x_n)

with some {{\tau_{ij}\in I}}. Let us prove this for {{n=2}}:

\displaystyle  \begin{vmatrix} f_1(x_1)&f_1(x_2)\\ f_2(x_1) & f_2(x_2) \end{vmatrix}= \begin{vmatrix} f_1(x_1)&f_1(x_2)-f_1(x_1)\\ f_2(x_1) & f_2(x_2)-f_2(x_1) \end{vmatrix} =\begin{vmatrix} f_1(\tau_{11})&f_1'(\tau_{12})\\ f_2(\tau_{21})&f_1'(\tau_{22}) \end{vmatrix} (x_2-x_1)

by the Lagrange theorem. (We take {{\tau_{11}=x_1}} and {{\tau_{21}=x_2}}.) The proof of the general case is pretty the same, but with Lagrange replaced by Schwarz.

If the interval {I} is compact, we obtain the following consequence: there exists a constant {C} (depending on the functions {{f_1, \ldots, f_n}}) such that for any {{x_1, \ldots, x_n\in I}}

\displaystyle   \Bigl|\det \bigl[f_i(x_j)\bigr]\Bigr|\le CV(x_1, \ldots, x_n)\le C \bigl(\max|x_i-x_j|\bigr)^{n(n-1)/2}. \ \ \ \ \ (1)

Now fix a positive integer {d}. In the sequel by a {d}-curve we mean a plane algebraic curve {{P(x,y)=0}} of degree less than {d} in {x} and in {y}. Then {d^2} points {{(x_1,y_1), \ldots, (x_{d^2}, y_{d^2})}} do not lie on a {d}-curve if and only if the determinant

\displaystyle  \Delta=\Delta \bigl((x_1,y_1), \ldots, (x_{d^2}, y_{d^2})\bigr)=\det \Bigl[x_k^iy_k^j\Bigr]_{\genfrac{}{}{0pt}{}{ 0\le i,j< d}{1\le k\le d^2}}

is non-zero. (Here {(i,j)} is the “vertical” index, and {k} is the “horizontal” index.) If all the coordinates {x_k} and {y_k} are rational numbers with denominator {N}, then {\Delta} is a non-zero rational number with denominator at most {N^{d^2(d-1)}}. We obtain the lower estimate

\displaystyle  |\Delta|\ge N^{-d^2(d-1)}.

Now let {f(x)} be a sufficiently smooth function on a compact interval {I} and {\Gamma} the plane curve {{y=f(x)}}. If all our points lie on {\Gamma} then {\Delta} can be expressed like {\det \bigl[g_{ij}(x_k)\bigr]} with {{g_{ij}(x)=x^if(x)^j}}. Applying inequality (1) with the functions {g_{ij}} instead of {f_i}, we obtain the following upper estimate:

\displaystyle  |\Delta|\le C \bigl(\max|x_i-x_j|\bigr)^{d^2(d^2-1)/2},

where {C} depends on {f} and {d}. Thus, if our points

  • have rational numbers with denominator {N} as coordinates,
  • lie on {\Gamma}, and
  • do not lie on a {d}-curve,


\displaystyle  \max|x_i-x_j|\ge \kappa N^{-2/(d+1)}, \qquad \kappa=\kappa(f,d)>0.

The same is true if we take more than {d^2} points with these properties, because we can always select among them {d^2} points not lying on a {d}-curve.

Splitting our interval into subintervals of length less than {\kappa N^{-2/(d+1)}}, we obtain the following statement (“the main lemma” of Bombieri and Pila):

the set {\Gamma \cap (N^{-1}{\mathbb Z})^2} lies on {O\bigl(N^{2/(d+1)}\bigr)}   {d}-curves.

We are ready to prove the theorem of Bombieri and Pila. Let {\Gamma} be a compact segment of a plane analytic transcendental curve. We may assume that {\Gamma} is of the type {y=f(x)}. Also, a compactness argument shows that number of intersections of {\Gamma} with any {d}-curve is bounded by a constant, depending on {\Gamma} and {d}. It follows that for any {d}

\displaystyle  \left|\Gamma \cap (N^{-1}{\mathbb Z})^2\right| \le c(\Gamma, d) N^{2/(d+1)}.

Selecting {d} small enough, we complete the proof.

(I thank the participants of the Groupe de travail “Géométrie diophantienne” in Bordeaux for their precious comments.)


6 Responses to “Schwarz mean value theorem and lattice points on analytic curves (after Bombieri and Pila)”

  1. Fedor Petrov Says:

    Thanks! In higher dimension it is less or more the same, yes?

  2. Sergei Yakovenko Says:

    Nice indeed!

    Is there anything similar to Shwarz formula for vector-valued functions? For functions of several variables?

  3. Lattice points on algebraic curves: a uniform estimate (after Bombieri and Pila) « Vieux Girondin Says:

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  4. A mean value theorem « Vieux Girondin Says:

    […] colleague asked me about the proof of the general mean value theorem mentioned in the very first entry of this blog. Let me recall the statement. In what follows is an interval and are (pairwise […]

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