AI 中文总结
研究有理系数齐次多项式向量场原点的全局渐近稳定性判定问题，通过构造保稳定性的可计算映射证明不同次数的全局渐近稳定性问题等价，结合归约方法得出其为多对一完全问题，进而证明该问题不可判定。
AI 中文摘要
我们研究有理系数齐次多项式向量场原点的全局渐近稳定性（GAS）判定问题。在本文第一部分，对每个固定维数$n$和奇数次数$2k+1$，我们给出一个可计算映射，该映射将$\mathbb{R}^n$上次数为$2k+1$的齐次向量场映射为$\mathbb{R}^m$、$m=\binom{n+k-1}{k}$上的齐次三次向量场，且同时保持GAS和李雅普诺夫稳定性。该映射引入新变量$y_\alpha=x^\alpha$（对应每个次数为$k$的单项式$x^\alpha$），在此变量下向量场变为三次型，并添加了垂直于形如$(x^\alpha)_{|\alpha|=k}$的点集的三次惩罚项；关键在于该惩罚项可控制系统的每个初始状态。由此可得，对每个固定奇数次数$d\ge3$，所有次数的GAS问题与三次次数的GAS问题两两多一等价。在第二部分，我们证明存在固定维数$N_*$和固定奇数次数$d_*$，使得$\mathbb{R}^{N_*}$上次数为$d_*$且原点为GAS的齐次向量场集合是多一完全的。该归约将整数未知量的不可判定多项式方程组族编码为雅可比θ级数乘积的常数项，把这些级数实现为有理系数多项式微分系统的输出，并将整数解的存在性转化为最大不变平均的符号，而齐次GAS可检测该符号。结合两部分结果可知，对每个奇数$d\ge3$和每个足够大的$N$，$\mathbb{R}^N$上次数为$d$的有理齐次向量场的GAS是多一完全问题；特别地，固定维数下三次向量场的GAS是不可判定的。基于近期研究工作，还证明了齐次向量场李雅普诺夫稳定性的若干类似结果。
英文摘要
We study the decision problem of global asymptotic stability (GAS) of the origin for homogeneous polynomial vector fields with rational coefficients. In the first part of the paper we give, for every fixed dimension $n$ and odd degree $2k+1$, a computable map which sends a homogeneous vector field of degree $2k+1$ on $\mathbb{R}^n$ to a homogeneous cubic vector field on $\mathbb{R}^m$, $m=\binom{n+k-1}{k}$, and preserves both GAS and Lyapunov stability. The map passes to new variables $y_α=x^α$ (one for each monomial $x^α$ of degree $k$), in which the vector field becomes cubic, and adds a cubic penalty term transverse to the set of points of the form $(x^α)_{|α|=k}$; the main point is that this controls every initial state of the new system. As a consequence, the GAS problems for all degrees, for each fixed odd degree $d\ge3$, and for degree three are pairwise many-one equivalent. In the second part we prove that there are a fixed dimension $N_*$ and a fixed odd degree $d_*$ such that the set of homogeneous vector fields on $\mathbb{R}^{N_*}$ of degree $d_*$ whose origin is GAS is many-one complete. The reduction encodes an undecidable family of polynomial equations in integer unknowns into the constant term of a product of Jacobi theta series, realizes these series as outputs of a polynomial differential system with rational coefficients, and turns the existence of an integer solution into the sign of a maximal invariant average, which homogeneous GAS detects. Combining the two parts shows that GAS of rational homogeneous vector fields on $\mathbb{R}^N$ of degree $d$ is many-one complete for every odd $d\ge3$ and every sufficiently large $N$; in particular it is undecidable for cubic vector fields in a fixed dimension. Building on recent work, some analogous results are proven for Lyapunov stability of homogeneous vector fields.
Comments50 pages
