In the Green-Tao method appear quite often expressions such as for some system of affine linear forms
where the
are some positive integers. The most basic example would be
where the last equality is here only to insist heavily on the form a general statement will have to take. A bit more generally, one easily sees that if
is d-periodic, then its average until
is basically its average until
. Following the ideas from Green-Tao, in fact we can show that for our general problem, the asymptotic should be analogously
where
is the gcd of the d’s.
Thus, applying this method, one has frequently to deal with such « local densities », the intersection of the zero sets of affine forms. More generally again, one may have to deal with things like for some polynomial
. Say
where the
are affine linear forms, so that
is of degree
. Suppose also that no two of them are affinely dependant, even when reduced modulo
. Then to compute
, notice that in order for
to satisfy
, either two of the forms must be divisible by
in n, or one must be divisible by
there.
To analyse the first possibility, reduce the forms modulo p, so they are as seen affine forms on the field . They are non-zero form, so the image of each linear part is of dimension
and the kernels of dimension
; so the zero locus of each of our affine forms is a hyperplane, and the hypothesis of no affine dependance ensures that the intersection of any two of these hyperplanes is of dimension
. So there are only
common zero for each pair of forms. But each point of
gives rise to
antecedents in
under the map of reduction modulo p. Finally, this possibility provides a summand of at most
to
.
To compute now , notice that any such 0 must arise from a 0 mod p. Now there are
0 mod p, because the set of zeros is a hyperplane. Then we can use a form of Hensel’s lemma in several variables. To obtain such a statement, we rely on the following simple identy for any
:
which shows that if , so
latex \text{grad}(Q)\neq 0\text{ mod }p$ then the vector
is constrained to live in a hyperplane if
is to vanish. So there are only
elements
with
and
. Of course if
is affine then the gradient is simply the linear part. Thus the number of roots modulo
is
. By the union bound, we see that:
. In the end,
.
To find more about this calculation and similar ones and the reason why I am interested in them, look at this article here.
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