Systems of affine equations, their L-functions, t-holonomic sequences, non-commutative determinants

Abstract

The authors defined in their previous paper ''$h^1\ne h_1$ for Anderson t-motives'' the notion of an affine equation associated to a t-motive. We study in the present paper general affine equations and their systems. We define L-functions of an affine equation. We show that its local factor is an algebraic function; we state a problem to prove an analog of the Lefschetz trace formula for the global L-function. We define t-holonomic sequences which are solutions of any affine equations. We state some conjectures and open questions related to t-holonomic sequences, for example formulas of their asymptotic growth. We show that some of the properties of t-holonomic sequences are analogous to the properties of the classical holonomic sequences, for example the closure properties. Further, we show that systems of affine equations can be reduced to one equation by a process analogous to the elimination of variables in a system of linear equations; this gives us a notion of non-commutative determinants. Finally, we consider some explicit examples of affine equations that come from Drinfeld modules, their duals and some other t-motives.

arXiv comment39 pages. New results are included