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

Abstract: We establish an effective Bertini-type theorem for hypersurfaces $X_f \colon f = 0$ defined over a finite field $k$ for which $f$ has no linear factors over the algebraic closure $\overline{k}$. Given a line $L$ defined over $k$ and a nonreduced $\overline{k}$-point $x$ on $X_f \cap L$, we give an upper bound on the number of planes $P$ containing $L$ for which $X_f \cap P$ 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 $K/\mathbb{Q}p$ is a finite extension with residue field isomorphic to $\mathbb{F}_q$ and $F(x_0, \ldots, x_n) \in K[x_0, \ldots, x{49}]$ 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.