Nonlinear Roth type theorems in finite fields

Bertini Theorems over Finite Fields

Let X be a smooth quasiprojective subscheme of P of dimension m ≥ 0 over Fq. Then there exist homogeneous polynomials f over Fq for which the intersection of X and the hypersurface f = 0 is smooth. In fact, the set of such f has a positive density, equal to ζX(m + 1) −1, where ζX(s) = ZX(q −s) is the zeta function of X. An analogue for regular quasiprojective schemes over Z is proved, assuming ...

Bertini Irreducibility Theorems over Finite Fields

Given a geometrically irreducible subscheme X ⊆ PnFq of dimension at least 2, we prove that the fraction of degree d hypersurfaces H such that H ∩X is geometrically irreducible tends to 1 as d→∞. We also prove variants in which X is over an extension of Fq, and in which the immersion X → PnFq is replaced by a more general morphism.

Extension Theorems in Vector Spaces over Finite Fields

Some Steinhaus type theorems over valued fields

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Bose-Burton type theorems for finite Grassmannians

In this paper both blocking sets with respect to the s-subspaces and covers with t-subspaces in a finite Grassmannian are investigated, especially focusing on geometric descriptions of blocking sets and covers of minimum size. When such a description exists, it is called a Bose–Burton type theorem. The canonical example of a blocking set with respect to the s-subspaces is the intersection of s ...