Affine $n$-space $A_k^n$ and its algebraic equivalent, the polynomial ring $k[x_1,…,x_n]$, are basic and widely studied objects in geometry and algebra, about which we know a great deal. However, there remains a host of basic open problems (like the Jacobian conjecture, Zariski Cancellation Conjecture, Complement Problem, …) indicating that our knowledge is nonetheless quite limited. In fact, the greatest obstacle in solving the above conjectures is our inability to “pinpoint” affine space among all varieties (or $k[x]$ among all finitely generated $k$-algebras): this is the so-called Characterization Problem.

The most recent approach to these problems is via additive group actions on affine $n$-space, which corresponds on the algebraic side, to the theory of locally nilpotent derivations. Using this, for instance, N. Gupta recently showed the falsitude of the Zariski Cancellation Conjecture in positive characteristic.

From a model-theoretic point of view, the polynomial ring (in its natural ring language) is quite expressive: in characteristic zero, one can define the integers (as a subset), one can express in general that, say, Embedded Resolution of Singularities holds, etc. Of course, one of the peculiarities of model theory (and probably one of the reasons for its pariah status) is the unavoidable presence of non-standard models. In other words, a characterization problem is never solvable in model theory, unless one allows some non first-order conditions as well (e.g., cardinality in categorical theories–but most mainstream mathematicians would not be too happy about that either). But other, more intrinsic problems arise: there are elementary equivalent fields whose polynomial rings are not. So, can we find an expanded language plus a “natural” but non first-order condition, that pinpoints the standard model, i.e., $k[x]$ within the models of its theory. Or even better, since these complete theories will have unwieldy axiomatizations, can we find a (recursive?) theory, whose only model satisfying the extra non first-order condition is $k[x]$?

In view of the recent developments in algebra/geometry, to this end, I will propose in this talk some languages that include additional sorts, in particularly, a sort for derivations. This is different from the usual language of differential fields, where one only studies a fixed (or possibly finitely many) derivation: we need all of them! We also need a substitute  for the notion of degree, and the corresponding group $Z$-action as power maps. To test our theories, we should verify which algebraic/geometric properties are reflected in this setup. For instance, affine $n$-space has no cohomology, which is equivalent to the exactness of the de Rham complex, and this latter statement is true in any of the proposed models. Nonetheless, this is only a preliminary analysis of the problem, and nothing too deep will yet be discussed in this talk.