Simplicial Differential Calculus, Divided Differences, and Construction of Weil Functors

TL;DR

This paper introduces a simplicial differential calculus that generalizes divided differences to broader contexts, offering a more efficient and characteristic-friendly approach to defining Weil functors via scalar extension.

Contribution

It develops a new simplicial calculus framework that simplifies evaluation point growth and extends the construction of Weil functors to positive characteristic settings.

Findings

Evaluation points grow linearly with degree

Applicable to topological modules over rings

Enables Weil functor construction in positive characteristic

Abstract

We define a simplicial differential calculus by generalizing divided differences from the case of curves to the case of general maps, defined on general topological vector spaces, or even on modules over a topological ring K. This calculus has the advantage that the number of evaluation points growths linearly with the degree, and not exponentially as in the classical, "cubic" approach. In particular, it is better adapted to the case of positive characteristic, where it permits to define Weil functors corresponding to scalar extension from K to truncated polynomial rings K[X]/(X^{k+1}).