# Counting Points

## Grothendieck group[edit | edit source]

A useful way to study the Grothendieck ring of varieties is via its realisation maps into a group , also known as *motivic measures* or *additive invariants*. These are maps from isomorphism classes of varieties that behave additive under disjoint unions, i.e. descend to . One such measure is counting points:

### Exercise[edit | edit source]

Complete this argument to show the claim about .

## Square polynomials[edit | edit source]

We can also tackle the first problem now and prove that assuming square values for all integers is itself the square of a polynomial in (although showing is also possible). For this, observe that is a square if and only if the hyperelliptic curve given by

Alternatively (or equivalently), one can use the Weil estimate for smooth, projective, geom. irreducible curves of genus :

### Exercise[edit | edit source]

Formulate and prove the most general version of this type of claim you can imagine. Possible answer: Let be a multivariate polynomial that assumes perfect power values at all integral points (possibly with different exponents). Then is a perfect power...