负相对余维数下的Thom级数
Thom series in negative relative codimension
AI总结：
本文建立接触函数奇点的Thom级数理论，证明其两项核心性质，阐明三类Thom多项式的关系并完成多类奇点的相关计算，成果将应用于计数几何问题研究。
AI中文摘要：
我们建立了接触函数奇点的Thom级数理论。接触奇点$\eta\in J^k(n,1)$的二次稳定化$\sigma_q$可生成接触奇点$\sigma_q^{i}\eta\in J^k(n+i,1)$。我们研究当$i$增大时，$\sigma_q^{i}\eta$的稳定Thom多项式（即商变量下的Thom多项式）的变化规律。为此，我们定义了接触函数奇点$\eta$的Thom级数，对任意给定的$i$，该级数是Schur多项式$s_\lambda$的线性组合。当$i$足够大时，Thom级数在$1-(n+i)$处的取值等于$\sigma_q^i\eta$的稳定Thom多项式；当$i$较小时，它可确定该多项式在特化映射核之外的系数。\n我们证明Thom级数具备两个核心性质：1）随着$i$增大，分拆$\lambda$遵循简单的稳定化模式，存在有限个$\lambda$的集合，通过该稳定化模式可生成整个Thom级数的支撑集；2）$s_\lambda$的系数是关于$i$的多项式，且有明确的次数上界。\n我们阐明了接触函数奇点的非稳定Thom多项式、稳定Thom多项式与Legendre Thom多项式之间的精确关系，并完成了三者的计算，具体如下：计算了$\gamma\leq6$的函数奇点的全部稳定Thom多项式族；计算了若干族二元与三元奇点的非稳定Thom多项式和Legendre Thom多项式；定义了一类多二元奇点，并计算了它们的Thom多项式；讨论了二阶Thom-Boardman类，并指出了函数奇点范畴之外会出现的难点。\n本文的研究结果将应用于一篇姊妹篇论文，届时Thom多项式将被用于解决计数几何中的问题。
英文摘要：
We develop a theory of Thom series for contact function singularities. Quadratic stabilization $σ_q$ of a contact singularity $η\in J^k(n,1)$ gives contact singularities $σ_q^{i}η\in J^k(n+i,1)$. We study the stable Thom polynomials (i.e.\ the Thom polynomial in quotient variables) of $σ_q^{i}η$ as $i$ increases. To this end, we define the Thom series of a contact function singularity $η$, which for any given $i$ is a linear combination of Schur polynomials $s_λ$. For large enough $i$ the value of the Thom series at $1-(n+i)$ is the stable Thom polynomial of $σ_q^iη$ and for small $i$ it determines its coefficients outside the kernel of a specialization map. We prove that the Thom series has two main properties: 1) as $i$ increases, the partitions $λ$ follow a simple stabilization pattern, and there is a finite set of $λ$ which generates the support of the entire Thom series via this stabilization; 2) the coefficients of the $s_λ$ are polynomials in $i$ with explicit degree bounds. We describe the precise relationship between unstable and stable Thom polynomials of contact function singularities and Legendre Thom polynomials and we carry out computations of all three, as follows. We compute the complete family of stable Thom polynomials for function singularities with $γ\leq 6$. We compute unstable and Legendre Thom polynomials for several families of binary and ternary singularities. We define a class of multi-binary singularities, and compute their Thom polynomials. We discuss second order Thom-Boardman classes, and indicate difficulties that arise beyond function singularities. The results of the paper will be used in a companion paper, where Thom polynomials will be applied to problems in enumerative geometry.
