发表机构
Kennesaw State University（肯尼索州立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文确定了正特征下初等对称多项式所有导数与多重性轨迹的维数，修正了Orzel猜想，并分类了达到最大维数的分支，证明其一般光滑，同时给出局部环、多重性及Frobenius正规形。
AI 中文摘要
我们确定了正特征代数闭域上初等对称多项式的所有导数轨迹和所有更高多重性轨迹的维数。对于n个变量中的d次多项式，设Q = p^{v_p(n-d+1)}。其q阶导数的公共零点轨迹的维数为d - q - 1 + 1_{Q>d-q}，而至少s重零点的轨迹的维数为d - s + 1_{Q>d}。二阶情形解决了Orzel猜想中的维数问题，并在d为p的幂时进行了必要的修正。我们分类了所有达到较大维数的分支，并证明它们一般是光滑的。在较小维数情形下，沿坐标分支，我们确定了局部环并精确得到一般多重性为Q。最后，我们识别了完整的平移稳定子概形并推导出Frobenius正规形。这些结果将维数跳跃与无穷小加厚区分开来，并统一描述了它们对特征的依赖。
英文摘要
We determine the dimensions of all derivative loci and all higher multiplicity loci of an elementary symmetric polynomial over an algebraically closed field of positive characteristic. For the degree-d polynomial in n variables, set Q = p^{v_p(n-d+1)}. The common zero locus of its order-q derivatives has dimension d - q - 1 + 1_{Q>d-q}, whereas the locus of zeros of multiplicity at least s has dimension d - s + 1_{Q>d}. The order-two case settles the dimension question in a conjecture of Orzel, with a necessary correction when d is a power of p. We classify all components attaining the larger dimension and prove that they are generically smooth. Along the coordinate components in the smaller-dimension cases, we determine the local rings and obtain generic multiplicity exactly Q. Finally, we identify the full translation stabilizer scheme and derive a Frobenius normal form. These results distinguish dimension jumps from infinitesimal thickening and give a uniform description of their dependence on the characteristic.
