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Let be a vector space over a finite prime field. One knows then which subsets
satisfy
, the sets that are stable under addition: these are the kernels of linear maps, that is, each such set
is the zero set of a bunch of linear forms.
Now we consider a two-coordinates analogous problem, that is, we consider a set and define the sets
and , V and H meaning vertical and horizontal. What are the sets that satisfy both
and
? Call such a set a transverse set. Natural exemples are cartesian products of vector spaces (which we shall call rectangles) as well as zero sets of bilinear forms. More generally, a set of the form
for some subspaces
and some bilinear forms
is horizontally and vertically closed. Call such a set a bilinear set, and call its codimensions the codimensions of the subspaces and the number of bilinear forms required (the dimension of the space of bilinear forms vanishing on it). What is the relationship between transverse sets and bilinear sets? Are all transverse sets essentially bilinear sets?
For a transverse set and
, let
be the fiber above x. Thus
and
is a vector space. Actually
depends only the projective class
. Moreover, the stability under horizontal operations is equivalent to the property that if
is on the projective line spanned by
and
, we have
.
The codimension 1 case
Suppose and each subspace
has codimension at most 1. Write
with
some vector uniquely defined up to homothety. It is easy to see that the set of
such that
is a vector subspace which we call
. Furthermore, if
, we have
, so that
descends to an injective map on
. Thus it is enough to study the case where
has codimension exactly one unless
. In other words, we have a map
satisfying the property that if
is on the projective line spanned by
and
, then
lies on the projective line spanned by
and
. Thus
maps projective lines into lines. Actually it can be quickly checked that such a map is either bijective or constant on each single projective line.
If we suppose it is injective, too, then by the fundamental theorem of projective geometry, we conclude that it is a projective map. Thus it is induced by a linear map . Hence writing
, we have shown that
is the zero set of the bilinear form
.
In general is far from being injective. It can for instance be constant on
(equal to
for some hyperplane
). In this case
. This is neither a rectangle, nor the zero form of a quadratic form, because if it was,
could be taken linear, and because
, it should be injective. In fact, if it is the intersection of zero sets of a family of bilinear forms, there needs to be
bilinear forms in this family, such as the forms
if (wlog)
is the hyperplane
. However
obviously contains a very large rectangle.
These are actually the only cases: such a map is either injective or constant. Indeed, if
is just a line, it’s obvious, so suppose
. Suppose that
maps projective lines into lines (in the sense above) but is neither injective nor constant. That is, there exist two distinct points
such that
, and a third point
satisfying
. This implies that
span a projective plane. Take a point
on the line spanned by
. Because
is a bijection on both lines
, you can find
on
such that
. Now consider the intersection
. Then you have
, so that on the line
, the map
is neither constant nor injective, which is absurd.
Arbitrary codimension
A more general potential counterexample, communicated to me by Ben Green, is a set of the form where
are sequences of increasing (resp. decreasing) subspaces. Thus for
, we have
where
. Again you can make it a bilinear set, but you need atrociously many bilinear forms.
However it contains a large rectangle. In fact, while
so that the density of
is at most
. Thus
. Finally by the pigeonhole principle, one of the cartesian products
have density at least
.
In general, we have a vector space . Suppose for instance each
has codimension 2. If we manage to find maps
such that
for each
, we are in good shape. Though it is not clear how to show that such a consistent choice of bases can be made…
The conjecture
If a has cardinality at least
, it contains a bilinear set of (linear and bilinear) codimensions
.
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