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 $x^m = 0$ 定义了直线上的一个脂肪点。该方案弧空间上的正则函数代数是 $k[x, x', x^{(2)}, \ldots]$ 除以 $x^m = 0$ 的所有微分推论所得的商。这一无限维代数通过对应于弧截断的有限维代数具有自然的滤过结构。我们证明其维数的生成级数等于 $\frac{m}{1 - mt}$。我们还确定了弧空间定义理想的词典序初始理想。这些结果受几何动机庞加莱级数的非约化版本、微分代数中的多重性以及弧空间与罗杰斯-拉马努金恒等式之间联系的启发。我们还证明了Afsharijoo最近在后者背景下提出的一个猜想。