The nullcone membership problem, deciding whether an orbit closure contains the origin, is fundamental in computational invariant theory. For self-adjoint groups, Bürgisser, Franks, Garg, Oliveira, Walter and Wigderson gave a geodesic optimization algorithm whose complexity is controlled by the gap, a condition number of the representation. We study the gap for quiver representations under the action of the special linear group. We prove that the inverse gap is polynomially bounded in the number of vertices and the maximum dimension for type A and $\hat{A}$, as well as tree quivers with uniform dimension vectors. Consequently, the algorithm of Bürgisser et al. solves the nullcone membership problem in polynomial time for these families. In contrast, we construct families of quivers and dimension vectors where the gap is exponentially small in the number of leaves, furthermore, for every connected quiver we exhibit dimension vectors such that the weight margin (a related condition number) is exponentially small in the number of vertices. We also extend our results to $σ$-semistability, thereby giving a new proof of a recent result of Iwamasa, Oki, and Soma.

翻译：零锥成员问题，即判断轨道闭包是否包含原点，是计算不变量理论中的基本问题。对于自伴群，Bürgisser、Franks、Garg、Oliveira、Walter 和 Wigderson 给出了一种测地线优化算法，其复杂度由该表示的间隙（条件数）控制。我们研究特殊线性群作用下箭图表示的间隙。我们证明，对于类型 A 和 $\hat{A}$ 以及具有均匀维数向量的树状箭图，逆间隙在顶点数和最大维数上具有多项式界。因此，Bürgisser 等人的算法可在多项式时间内解决这些族类的零锥成员问题。相反，我们构造了箭图和维数向量的族类，其中间隙在叶子数上呈指数级缩小；此外，对于每个连通箭图，我们展示了使得权重边际（一个相关的条件数）在顶点数上呈指数级缩小的维数向量。我们还将结果推广到 $\sigma$-半稳定性，从而给出 Iwamasa、Oki 和 Soma 近期结果的一个新证明。