AI 中文总结
该研究通过循环矩阵-向量对的模空间与余轨道映射，推导主理想码MDS轨迹的Schur-Plücker几何结构，给出通用MDS多项式，得到边界、滤过、坏特征准则等结果，并证明特定条件下第一失效层含非GRS稠密开轨迹。
AI 中文摘要
设\ng\n为次数\nr<n\n的首一多项式，\nC_g(n)\n为由满足\ndeg(ug)<n\n的倍式\nug\n构成的系数向量码。我们研究系数空间MDS轨迹\nM_{n,r}\n。伴随构造将系数空间与循环矩阵-向量对的模空间等同，余轨道映射将其嵌入为Gr(r,n)的标准大胞腔中的光滑完全交。我们证明，每个归一化极大Plücker坐标（差一符号）拉回至常数系数\nA_0\n的幂乘以Schur多项式\nS_κ(g)=s_κ(Λ_g)\n，其中\nκ⊆(n−r)^{r−1}\n。因此通用MDS多项式为\nD_{n,r}=A_0∏_{κ⊆(n−r)^{r−1}}S_κ\n。该描述给出了Z上的平坦非MDS边界、显式次数与有限域估计、由稀疏倍式决定的长度滤过，还给出了根重数层的坏特征准则及不可约层的密度1结果。最后，对r≥3且N≥r+3，代数闭域上每个非空的第一失效层都含稠密开非GRS轨迹。
英文摘要
Let \(g\) be a monic polynomial of degree \(r<n\), and let \(C_g(n)\) be the coefficient-vector code formed by multiples \(ug\) with \(°(ug)<n\). We study the coefficient-space MDS locus \(M_{n,r}\). The companion construction identifies coefficient space with the moduli of cyclic matrix-vector pairs, and the remainder-orbit map embeds it as a smooth complete intersection in the standard big cell of \(\operatorname{Gr}(r,n)\). We prove that every normalized maximal Plucker coordinate pulls back, up to sign, to a power of the constant coefficient \(A_0\) times a Schur polynomial \(S_κ(g)=s_κ(Λ_g)\), where \(κ\subseteq (n-r)^{r-1}\). Hence the universal MDS polynomial is \[D_{n,r}=A_0\prod_{κ\subseteq (n-r)^{r-1}}S_κ.\] This description yields a flat non-MDS boundary over \(\mathbb{Z}\), explicit degree and finite-field estimates, and a length filtration governed by sparse multiples. It also gives bad-characteristic criteria on root-multiplicity strata and density-one results on the irreducible stratum. Finally, for \(r\ge 3\) and \(N\ge r+3\), every nonempty first-failure layer over an algebraically closed field has a dense open non-GRS locus.
Comments48 pages
