Faster computation of Witt vectors over polynomial rings

TL;DR

The paper addresses efficient computation of the ring laws for Witt vectors of length $n$ over a polynomial ring with coefficients in a finite field. It uses Illusie’s isomorphism to move computations to a polynomial-ring setting via an injective map $\widetilde{F}^{n-1}$ and implements the approach in SageMath, achieving speedups over Finotti’s method. It also presents a second algorithm by Caruso (the phantom method) and a framework to extend to rings of mixed characteristic and to quotient rings, with detailed performance comparisons. The results provide a practical toolkit for arithmetic-geometry computations, p-adic cohomology, and cryptography where Witt vectors with polynomial coefficients arise, and clarify trade-offs between speed and memory across parameter regimes.

Abstract

We describe an algorithm which computes the ring laws for Witt vectors of finite length over a polynomial ring with coefficients in a finite field. This algorithm uses an isomorphism of Illusie in order to compute in an adequate polynomial ring. We also give an implementation of the algorithm in SageMath, which turns out to be faster that Finotti's algorithm, which was until now the most efficient one for these operations.

Figures (5)

• Figure 1: Comparisons in $W_5\mleft(\mathbb F_{3^{10}}\mleft[X\mright]\mright)$.
• Figure 2: Comparisons in $W_n\mleft(\mathbb F_{5^2}\mleft[X\mright]\mright)$ with $d=2$.
• Figure 3: Comparisons in $W_4\mleft(\mathbb F_p\mleft[X\mright]\mright)$ with $d=10$.
• Figure 4: Comparisons in $W_4\mleft(\mathbb F_q\mleft[X\mright]\mright)$ with $d=5$ and $p=7$.
• Figure 5: Comparisons in $W_4\mleft(\mathbb F_q\mleft[X,Y\mright]\mright)$ with total degrees of the coefficients at most $2$.

Theorems & Definitions (2)

• proof
• proof