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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.
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