By Henri Linus Breloer – April 2026
Hankel matrices arise in a variety of contexts. They play a key role in statistics and tensor methods. From the perspective of algebraic geometry, the space of Hankel matrices with rank no greater than some fixed r forms an affine variety. These spaces are well-structured and can be effectively studied using classical tools from commutative algebra and algebraic geometry. However, when generalized to infinite Hankel matrices of finite rank no greater than r, the situation becomes more complex, as these objects do not align as neatly with standard methods.
The coordinate ring of the space of infinite Hankel matrices is a non-Noetherian graded ring, defined as a quotient of the polynomial ring in infinitely many variables, where the degree of the n-th variable is n. Despite its infinite nature, this structure allows us to define a Hilbert series for the ring.
Remarkably, the Hilbert series for this space remains a rational function, mirroring the behavior of graded finitely generated algebras over a field K. This surprising result connects to a deeper theory, which we explore in detail in our upcoming paper. Starting from the case of arbitrary ideals in the infinite polynomial ring with usual grading, where a Hilbert series is generally ill defined, we restrict to ideals invariant under permutation of variables. Analyzing this underlying symmetry allows us to find rationality results just like in the finite case.