Let $F$ be a complete discretely valued field with ring of valuation $R$, uniformizer $\pi$, and residue characteristic $\neq 2$. A theorem of Springer says that a quadratic form $q=q_1 \bot \pi q_2$, where $q_1,q_2$ have coefficients in $R^\times$, is isotropic over $F$ if and only if at least one of the images of $q_1,q_2$ is isotropic over the residue field $R/\pi$.
I am interested in knowing whether:
a similar result holds for other varieties than quadrics (this is a bit vague, but anything that comes to mind might be useful),
can anything remotely similar be said for quadratic forms in the dyadic case (i.e. when char $R/\pi=2$)?
(Note: I first asked the question on mathoverflow, but did not get an answer. That post is now deleted.)
For Q2, the answer is yes. See Section 19 of " the algebraic and geometric theory of quadratic forms", Elman, Karpenko, Merkurjev, AMS Coll. Pub.56,2008.
For Q1: You can generalize the fact that if $q_1,q_2$ are anistropic modulo $\pi$, then $q$ is anisotropic to homogenous forms of degree $d$.
What will be missing is: if a homogeneous form $f$ (with the appropriate notion of nondegeneracy, which is not clear to me) of degree $d\geq 2$ has no nontrivial zeroes, then it has no nontrivial zeroes modulo $\pi$.
I don't know if it is reasonably true (i suspect it isn't).