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 $k[x,x^{\prime },x^{(2)},\dots <!--FUNCTION\; APPLICATION-->]$ 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 $m∕(1-mt)$. We also determine the lexicographic initial ideal of the defining ideal of the arc space. These results are motivated by the nonreduced version of the geometric motivic Poincaré 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.

等式$X^{米}=0$定义线上的胖点。该方案的弧空间上的正则函数的代数是$k[X,X^{\prime },X^{（2）},\mathrm{\dots \dots }<!--FUNCTION\; APPLICATION-->]$由所有不同的后果$X^{米}=0$。这个无限维代数允许通过对应于弧截断的有限维代数进行自然过滤。我们证明它们的维数的生成级数等于$米∕（1-米t）$。我们还确定了弧空间定义理想的词典初始理想。这些结果是由几何动机庞加莱级数的非约化版本、微分代数的多重性以及弧空间和罗杰斯-拉马努金恒等式之间的联系所激发的。我们还证明了 Afsharijoo 最近在后一种情况下提出的猜想。