Take $\mathbb{Z}^n$ equipped with two symmetric bilinear forms, one positive-definite $(\cdot,\cdot)_A : \mathbb{Z}^n \times \mathbb{Z}^n \to \mathbb{R}$ and one indefinite $(\cdot,\cdot)_J : \mathbb{Z}^n \times \mathbb{Z}^n \to \mathbb{Z}$. Let $A$ and $J$ denote the associated Gram matrices in the standard basis for $\mathbb{Z}^n$. Assume that both $A$ and $J$ have determinant $1$. Take $J$ to be a $p,q$ signature matrix, with $n = p+q$.
Define the set of nonzero null vectors $N_J \equiv \{ v \in \mathbb{Z}^n \setminus \{0\} : (v,v)_J =0\}$ and let $$ \mu_0(A) = \min_{v \in N_J} \ (v,v)_A $$ Question: What can be said about this "null SVP". Are there algorithms I can use to compute $\mu_0(A)$, and to find the associated shortest null vector $v_0$? Is there a decent upper bound on $\mu_0(A)$ as a function of dimension $n$ (and probably the signature $p,q$)?
If we ignore $J$, then $A$ defines some positive definite lattice of dimension $n$. If $n$ is not too large, there are algorithms (LLL, etc) that I can use to find the shortest nonzero vectors of $A$, and to compute their norm $\mu(A)$. Even without doing any computation, I have at my disposal decent upper bounds for $\mu(A)$: the Hermite constant in dimensions where it is known, and results like Rogers' upper bound for sphere packing density in other dimensions. I am hoping that some analogous results are available even when the restriction to null vectors is enforced.
At the moment, the only method I can think of for computing $\mu_0(A)$ involves enumerating short vectors in the even sublattice $D_n$ of $\mathbb{Z}^n$, and then iteratively eliminating the shortest ones until I get to a null vector. The restriction to $D_n$ is because I can split any null vector as $v = v_+ + v_-$, with $(v_+,v_+)_J = - (v_-,v_-)_J > 0$. However, $D_n$ is also the smallest sublattice that contains all null vectors, so I cannot push this method any further.
