We consider a variant of the mortality problem: given $k\times k$ matrices $A_1,\dots ,A_t$, do there exist nonnegative integers $m_1,\dots ,m_t$ such that $A_1^{m_1}\cdots A_t^{m_t}$ equals the zero matrix? This problem is known to be decidable when $t\le 2$ but undecidable for integer matrices with sufficiently large t and k.

We prove that for $t=3$ this problem is Turing-equivalent to Skolem's problem and thus decidable for $k\le 3$ (resp. $k=4$) over (resp. real) algebraic numbers. Consequently, the set of triples $(m_1,m_2,m_3)$ for which the equation $A_1^{m_1}{A}_2^{m_2}{A}_3^{m_3}$ equals the zero matrix is a finite union of direct products of semilinear sets.

For $t=4$ we show that the solution set can be non-semilinear, and thus there is unlikely to be a connection to Skolem's problem. We prove decidability for upper-triangular $2\times 2$ rational matrices by employing powerful tools from transcendence theory such as Baker's theorem and S-unit equations.

我们考虑死亡率问题的一个变体：给定 $克\times 克$ 矩阵 ${\mathrm{一种}}_1,\dots ,{\mathrm{一种}}_{吨}$, 是否存在非负整数 ${米}_1,\dots ,{米}_{吨}$ 以至于 ${\mathrm{一种}}_1^{{米}_1}\cdots {\mathrm{一种}}_{吨}^{{米}_{吨}}$等于零矩阵？已知这个问题是可判定的，当$吨\le 2$但对于具有足够大t和k 的整数矩阵是不可判定的。

我们证明对于 $吨=3$ 这个问题与 Skolem 问题是图灵等价的，因此可判定为 $克\le 3$ （分别 $克=4$) 超过（或实数）代数数。因此，三元组$({米}_1,{米}_2,{米}_3)$ 其中方程 ${\mathrm{一种}}_1^{{米}_1}{\mathrm{一种}}_2^{{米}_2}{\mathrm{一种}}_3^{{米}_3}$ 等于零矩阵是半线性集的直接积的有限并集。

为了 $吨=4$我们表明解决方案集可以是非半线性的，因此不太可能与 Skolem 问题有联系。我们证明了上三角的可判定性$2\times 2$ 通过使用超越理论中的强大工具（例如贝克定理和 S 单位方程）来创建有理矩阵。