Effective Bertini theorems and zeros of p-adic forms of degree 7

We establish an effective Bertini-type theorem for hypersurfaces Xf f = 0 defined over a finite field k for which f has no linear factors over the algebraic closure k̄. Given a line L defined over k and a nonreduced k̄-point x on Xf ∩ L, we give an upper bound on the number of planes P containing L for which Xf ∩ P contains a line through x. Underlying this result is a factorization algorithm for bivariate polynomials originally due to Kaltofen, which we present with slightly relaxed hypotheses. Our primary application is to Artin's conjecture on p-adic forms of degree 7: if K/Qₚ is a finite extension with residue field isomorphic to Fq and F(x₀, …, xₙ) ∈ K[x₀, …, x₄₉] is a homogeneous form of degree 7, then there exists a K-solution to $F=0$ whenever $q>679$. This improves on a result of Wooley.