对于给定的双线性系统，其增长率是否为1是不可判定的
It is undecidable whether the growth rate of a given bilinear system is 1
摘要：
我们发现，有没有算法，决定任何双线性系统$(B,v)$，如果增长率的$(B,v)$是$1$。这回答了Bui的一个问题，他表明，如果系数为正，则增长率是可计算的（即，有一种算法输出$(B,v)$的增长率的数字序列）。我们的证明是基于一组矩阵的联合谱半径的计算的双线性系统的增长率的计算减少。我们还使用我们的减少推断，有没有算法，近似的增长率的双线性系统的相对精度$\varepsilon$在时间多项式的系统和$\varepsilon$的大小。即使所有系数都是非负有理数，我们的两个结果也成立。
英文摘要：
We show that there exists no algorithm that decides for any bilinear system $(B,v)$ if the growth rate of $(B,v)$ is $1$. This answers a question of Bui who showed that if the coefficients are positive the growth rate is computable (i.e., there is an algorithm that outputs the sequence of digits of the growth rate of $(B,v)$). Our proof is based on a reduction of the computation of the joint spectral radius of a set of matrices to the computation of the growth rate of a bilinear system. We also use our reduction to deduce that there exists no algorithm that approximates the growth rate of a bilinear system with relative accuracy $\varepsilon$ in time polynomial in the size of the system and of $\varepsilon$. Our two results hold even if all the coefficients are nonnegative rationals.
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