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 k. Given a line L defined over k and a nonreduced k-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) Kx₀, , 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.
Beneish et al. (Wed,) studied this question.