The equation $x^m = 0$ defines a fat point on a line. The algebra of regular functions on the arc space of this scheme is the quotient of $k[x, x', x^{(2)}, \ldots]$ by all differential consequences of $x^m = 0$. This infinite-dimensional algebra admits a natural filtration by finite dimensional algebras corresponding to the truncations of arcs. We show that the generating series for their dimensions equals $\frac{m}{1 - mt}$. We also determine the lexicographic initial ideal of the defining ideal of the arc space. These results are motivated by nonreduced version of the geometric motivic Poincar\'e series, multiplicities in differential algebra, and connections between arc spaces and the Rogers-Ramanujan identities. We also prove a recent conjecture put forth by Afsharijoo in the latter context.

翻译：方程式 $x ⁇ m = 0 = 0 定义线上的一个脂肪点。 这个图案的弧空间上正常函数的代数是 $k[x, x', x ⁇ (2)},\ ldots]$的所有不同后果的商数 $x ⁇ m = 0 美元。 这个无限的代数承认了与弧轨迹相对应的有限维维代数的自然过滤。 我们显示, 其尺寸的生成序列等于 $\ frac{m ⁇ 1 - mt} 。 我们还确定了弧空间定义理想的字典初步理想。 这些结果的动机是: 数位模型的不简化版本 Poincar\ e 系列, 差位数的多位数, 弧空间与罗杰斯- 曼尼扬 身份的连接。 我们还证明了 Afsharijoo 在后一种背景下最近绘制的直径。