Counting vanishing matrix-vector products

Consider the following parameterized counting variation of the classic subset sum problem, which arises notably in the context of higher homotopy groups of topological spaces. Let $v\in Q^d$ be a rational vector, $(T_1,T_2\dots ,T_m)$ a list of $d\times d$ rational matrices, $S\in Q^{h\times d}$ a rational matrix not necessarily square and k a parameter. The goal is to compute the number of ways one can choose k matrices $T_{i_1},T_{i_2},\dots ,T_{i_k}$ from the list such that $S{T}_{i_k}\cdots T_{i_1}v=0\in Q^h$.

In this paper, we show that this problem is $\#W\left[2\right]$-hard for parameter k. As a consequence, computing the k-th homotopy group of a d-dimensional 1-connected topological space for $d>3$ is $\#W\left[2\right]$-hard for parameter k. We also discuss a decision version of the problem and its several modifications for which we show $W\left[1\right]/W\left[2\right]$-hardness. This is in contrast to the parameterized k-sum problem, which is only $W\left[1\right]$-hard (Abboud-Lewi-Williams, ESA'14). In addition, we show that the decision version of the problem without parameter is an undecidable problem, and we give a fixed-parameter tractable algorithm for matrices of bounded size over finite fields, parameterized by the matrix dimensions and the order of the field.