Stability of ranks under field extensions

Stability of ranks under field extensions, Discrete Analysis 2025:27, 29 pp.

There are several notions of complexity associated with tensors, each capturing different aspects of their algebraic or geometric structure. These notions depend sensitively on the field over which the tensors are defined, making it important to understand how they behave under field extensions. The main focus of this paper is to study the relationships between the different notions of rank and to investigate their behavior under field extension.

The first notion studied is that of analytic rank, defined for tensors over finite fields and first introduced by Gowers and Wolf. This notion measures a type of complexity associated with the equidistribution properties of the tensor. The authors show that the analytic rank can change by at most a multiplicative constant factor—depending only on the degree of the tensor and the base field—when the field is extended.

Another important notion of complexity is that of partition rank, the tensor analogue of a concept introduced by Schmidt in his study of integer points on homogeneous varieties. Here, complexity is measured in an algebraic sense, through the minimal length of a representation of the tensor as a sum of products of tensors of lower degree. Two central conjectures concern the partition rank: the first, a stability conjecture, asserts that the partition rank can decrease by at most a multiplicative constant factor under field extensions; the second posits that the partition rank and analytic rank over finite fields differ by at most a multiplicative constant factor. The authors show that these two conjectures are, in fact, equivalent.

The paper also investigates other important notions of tensor complexity, such as geometric rank and slice rank, and explores the relations between these different measures. Precise definitions, detailed proofs, and further discussion of the connections among these notions can be found in the paper.