Let be a compact segment of a plane real analytic transcendental curve. Bombieri and Pila (1989) proved that may not contain “many” rational points with a given denominator. Precisely: given , there exists a constant such that for any positive integer

(As they remark, the same statement holds if 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 , a function , and points , there exists a point such that

where is the Vandermond determinant. The case 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 be an interval, and . Then

with some . Let us prove this for :

by the Lagrange theorem. (We take and .) The proof of the general case is pretty the same, but with Lagrange replaced by Schwarz.

If the interval is compact, we obtain the following consequence: there exists a constant (depending on the functions ) such that for any

Now fix a positive integer . In the sequel by a *-curve* we mean a plane algebraic curve of degree less than in and in . Then points do **not** lie on a -curve if and only if the determinant

is non-zero. (Here is the “vertical” index, and is the “horizontal” index.) If all the coordinates and are rational numbers with denominator , then is a non-zero rational number with denominator at most . We obtain the **lower** estimate

Now let be a sufficiently smooth function on a compact interval and the plane curve . If all our points lie on then can be expressed like with . Applying inequality (1) with the functions instead of , we obtain the following **upper** estimate:

where depends on and . Thus, if our points

- have rational numbers with denominator as coordinates,
- lie on , and
- do not lie on a -curve,

then

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

Splitting our interval into subintervals of length less than , we obtain the following statement (“the main lemma” of Bombieri and Pila):

the set lies on -curves.

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

Selecting small enough, we complete the proof.

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

April 4, 2010 at 4:28 am |

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

April 4, 2010 at 4:30 am |

I believe so, but this is not straightforward.

April 12, 2010 at 9:41 am |

Nice indeed!

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

April 12, 2010 at 12:04 pm |

No idea. I believe it should have similar generalizations as Lagrange, but I learn these things only when I need them.

April 23, 2010 at 2:54 am |

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