We study the algebraic K-theory of rings of the form \(R[x]/x^e\) . We do this via trace methods and filtrations on topological Hochschild homology and related theories by quasisyntomic sheaves. We produce computations for R a perfectoid ring in terms of the big Witt vectors of R, for R a smooth curve over a perfectoid ring in terms of the prismatic cohomology of R, and for R a complete mixed characteristic discrete valuation rings with perfect residue field in terms of the prismatic cohomology and Hodge–Tate divisor of R.
