论文标题
通用LEX扩展希尔伯特功能和汉密尔顿的理想近似
Universal Lex Ideal Approximations of Extended Hilbert Functions and Hamilton Numbers
论文作者
论文摘要
令$ r^h $表示变量中的多项式环$ x_1，\，\ ldots，\，x_h $在指定字段$ k $上。我们同时考虑所有这些戒指，在每种使用词典（LEX）单一订单中，带有$ x_1> \ cdots> x_h $。鉴于固定的均匀理想$ i $ in $ r^h $，对于每$ d $，最多都有$ d $产生的独特的lex理想，其Hilbert函数与Hilbert功能符合$ i $ $ $ $最dent $ d $的Hilbert功能。当我们考虑$ ir^n $ for $ n \ geq h $时，set $ \ mathfrak {b} _d（i，n）$的最小发电机的最小发电机最多可以更改，但是$ d $可能会更改，但是$ \ m athfrak {b} _d（i，n）$是所有$ n \ gg gg gg gg gg 0 $ ns formand os n \ gg gg gg 0 $。我们让$ \ mathfrak {b} _d（i）$表示所有$ n \ gg 0 $获得的发电机集，我们让$ b_d = b_d（i）$是其基数。 $ b_1，\，\ ldots，\，b_d，\，\ ldots $以这种方式获得的序列可能会非常快。值得注意的是，即使$ i =（x_1^2，x_2^2）$，也获得了一个非常有趣的序列，0、2、3、4、6、12、924、409620，$ \，\ ldots $。此序列与$ h_ {d-1} + 1 $ for $ d \ geq 2 $相同，其中$ h_d $是$ d \，$ th hamilton编号。汉密尔顿和哈蒙德和西尔维斯特研究了汉密尔顿的数字，因为它们在操纵多项式方程中使用Tschirnhaus转换有关的计数问题。
Let $R^h$ denote the polynomial ring in variables $x_1,\,\ldots,\, x_h$ over a specified field $K$. We consider all of these rings simultaneously, and in each use lexicographic (lex) monomial order with $x_1 > \cdots > x_h$. Given a fixed homogeneous ideal $I$ in $R^h$, for each $d$ there is unique lex ideal generated in degree at most $d$ whose Hilbert function agrees with the Hilbert function of $I$ up to degree $d$. When we consider $IR^N$ for $N \geq h$, the set $\mathfrak{B}_d(I,N)$ of minimal generators for this lex ideal in degree at most $d$ may change, but $\mathfrak{B}_d(I,N)$ is constant for all $N \gg 0$. We let $\mathfrak{B}_d(I)$ denote the set of generators one obtains for all $N \gg 0$, and we let $b_d = b_d(I)$ be its cardinality. The sequences $b_1, \, \ldots, \, b_d, \, \ldots$ obtained in this way may grow very fast. Remarkably, even when $I = (x_1^2, x_2^2)$, one obtains a very interesting sequence, 0, 2, 3, 4, 6, 12, 924, 409620,$\,\ldots$. This sequence is the same as $H_{d-1} + 1$ for $d \geq 2$, where $H_d$ is the $d\,$th Hamilton number. The Hamilton numbers were studied by Hamilton and by Hammond and Sylvester because of their occurrence in a counting problem connected with the use of Tschirnhaus transformations in manipulating polynomial equations.
