AI 中文总结
研究循环算子\(A\)、循环向量\(z\)的Krylov码的MDS轨道段与有理正规曲线的关系，证明特定共轭条件下二者的联系，给出有限域和代数闭域上相关分类及性质，还涉及多项式数量公式、轨迹定义及比例结果。
AI 中文摘要
设\(A\)是域\(k\)上\(r\)维向量空间上的循环算子，\(z\)是循环向量，其Krylov码具有校验矩阵\((z,Az,\ldots,A^{n - 1}z)\)。对于\(r\geq3\)且\(n\geq r + 3\)，证明当射影对\((A,[z])\)与由\(\mathrm{PGL}_2\)的\((r - 1)\)次对称幂作用产生的射影对共轭时，MDS轨道段恰好在有理正规曲线上。在有限域上，对于伴随算子，给出广义Reed - Solomon轨迹的分裂半单、两个非分裂半单和幂幺族的完整分类。在代数闭域\(k\)上，半单GRS系数轨迹的扎里斯基闭包\(\GRSsurf_{r,k}\)是不可约有理曲面，一般由几何级数根集的二维环面双射参数化；反转是一般模糊性。参数环面通过反转的仿射商是\(\GRSsurf_{r,k}\cap D(a_0)\)（其非零常数项开部分）的正规化。在首一\(r\)次多项式空间中的余维数为\(r - 2\)。弗罗贝尼乌斯下降给出\(\mathbb{F}_q\)上GRS多项式数量的精确公式。规范剩余校验矩阵通过主开条件定义MDS轨迹。对于固定的\(r\geq3\)和\(n\geq r + 3\)，当\(q\)通过素数幂趋于无穷时，\(\mathbb{F}_q\)上其伴随码为MDS且非GRS的所有首一\(r\)次多项式的比例趋于\(1\)。
英文摘要
Let \(A\) be a cyclic operator on an \(r\)-dimensional vector space over a field \(k\), and let \(z\) be a cyclic vector. Their Krylov code has parity-check matrix \((z,Az,\ldots,A^{n-1}z)\). For \(r\ge 3\) and \(n\ge r+3\), we prove that an MDS orbit segment lies on a rational normal curve precisely when the projective pair \((A,[z])\) is conjugate to one arising from the \((r-1)\)-st symmetric-power action of \(\mathrm{PGL}_2\). Over finite fields, for companion operators, this gives a complete classification of the generalized Reed--Solomon locus into split semisimple, two nonsplit semisimple, and unipotent families. Over an algebraically closed field \(k\), the Zariski closure \(\GRSsurf_{r,k}\) of the semisimple GRS coefficient locus is an irreducible rational surface, generically parameterized two-to-one by a two-dimensional torus of geometric-progression root sets; reversal is the generic ambiguity. The affine quotient of the parameter torus by reversal is the normalization of \(\GRSsurf_{r,k}\cap D(a_0)\), its nonzero-constant-term open part. The codimension in the space of monic degree-\(r\) polynomials is \(r-2\). Frobenius descent gives an exact formula for the number of GRS polynomials over \(\mathbb F_q\). A canonical remainder parity-check matrix defines the MDS locus by a principal open condition. For fixed \(r\ge3\) and \(n\ge r+3\), the proportion of all monic degree-\(r\) polynomials over \(\mathbb F_q\) whose companion codes are MDS and non-GRS tends to one as \(q\to\infty\) through prime powers.
Comments39 pages
