Simplicial differential calculus, divided differences, and construction of Weil functors

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 grows 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 ).2000 Mathematics Subject Classification. 13B02, 13F20, 13N99, 14B10, 39A12, 58A20, 58A32.

Lie Calculus

“…However, the general Taylor formula from [BGN04] does make sense over any base ring. A closer inspection shows that this formula really belongs to full cubic calculus, and more precisely to the "non-symmetric" aspect of full calculus, which has been christianed in [Be13] simplicial differential calculus. Thus, although symmetric cubic calculus can be defined over any base ring, it is sort of "incomplete" in certain cases (such as finite rings).…”

“…When working on the foundations of differential calculus (in chronological order, [BGN04,Be08,Be13,Be15a,Be15b]), I got the impression that there ought to exist a comprehensive algebraic theory, englobing both the fundamental results of calculus and of differential geometry, and where Lie theory is a kind of Ariadne's thread. Confirming this impression, groupoids turned out, in my most recent approach [Be15a,Be15b], to be the most remarkable algebraic structure underlying calculus.…”

Higher order extension of Löwner’s theory: Operator 𝑘-tone functions

“…In particular, for α, β ∈ (a, b) with α < β and for distinct t 0 , t 1 , · · · ∈ [0, 1] let x k := (1 − t k )α + t k β; we then notice that [7].) Lemma 1.1.…”

Conceptual differential calculus part ii: Cubic higher order calculus

“…In particular, there is a natural action of the symmetric group S n preserving the structure ("Schwarz' theorem"), whence the term symmetric cubic calculus. However, for proving the general Taylor formula from [BGN04], the symmetric cubic setting is not sufficient, and for this reason we have developed in [Be13] a simplicial differential calculus. It should be possible and important (in particular, in order to relate the present work to approaches used in algebraic geometry) to develop a conceptual and categorical formulation of simplicial calculus -however, to keep the present work in reasonable bounds, this is left for subsequent work.…”

“…Simplicial calculus. In [Be13], I take as departure point an "explicit formula" which can be seen as a simplicial analog of the explicit formula from Theorem 2.3. We shall give a conceptual interpretation in subsequent work.…”