$h^1$, $h_1$ of Anderson t-motives, systems of affine equations and non-commutative determinants

Abstract

The authors defined in "$h^1\ne h_1$ for Anderson t-motives" the notion of an affine equation associated to a t-motive $M$. Here we define two systems of affine equations associated to a t-motive $M$, used for calculation of $H^1(M)$ and $H_1(M)$. We describe the process of elimination of unknowns in these systems. This is an analog of the corresponding theory of systems of linear differential equations. It gives us a notion of a non-commutative determinant $det_{i,c}(M)$ which belongs to the Anderson ring $\Bbb C_\infty[T,\tau]$ of non-commutative polynomials. Finally, we calculate $det_{i,c}(M)$ for $M=$ a Drinfeld module or its 1-dual. Also, some explicit calculations are made for Anderson t-motives of dimension $n$, rank $2n$. Some problems of future research are formulated.