Related papers: Gotzmann's persistence theorem for finite modules

Gotzmann's Persistence states that the growth of an arbitrary ideal can be controlled by comparing it to the growth of the lexicographic ideal. This is used, for instance, in finding equations which cut out the Hilbert scheme (of subschemes…

Gotzmann proved the persistence for minimal growth for ideals. His theorem is called Gotzmann's persistence theorem. In this paper, based on the combinatorics on binomial coefficients, a simple combinatorial proof of Gotzmann's persistence…

Gotzmann's persistence theorem provides a method for determining the Hilbert polynomial of a subscheme of projective space by evaluating the Hilbert function at only two points, irrespective of the dimension of the ambient space. In…

We establish a form of the Gotzmann representation of the Hilbert polynomial based on rank and generating degrees of a module, which allow for a generalization of Gotzmann's Regularity Theorem. Under an additional assumption on the…

Gotzmann's persistence theorem enables us to confirm the Hilbert polynomial of a subscheme of projective space by checking the Hilbert function in just two points, regardless of the dimension of the ambient space. We generalise this result…

In recent work, generalized persistence modules have proved useful in distinguishing noise from the legitimate topological features of a data set. Algebraically, generalized persistence modules can be viewed as representations for the poset…

We present a generalization of the induced matching theorem and use it to prove a generalization of the algebraic stability theorem for $\mathbb{R}$-indexed pointwise finite-dimensional persistence modules. Via numerous examples, we show…

A Gotzmann monomial ideal of the polynomial ring is a monomial ideal which is generated in one degree and which satisfies Gotzmann's persistence theorem. A subset $V$ is said to be a Gotzmann subset if the ideal generated by $V$ is a…

Let $\mathbf{k}$ be a field and let $V: \mathscr{C} \to \mathbf{k}\textup{-Mod}$ be a point-wise finite dimensional persistence modules, where $\mathscr{C}$ is a small category. Assume that for all local Artinian $\mathbf{k}$-algebras $R$…

In this paper, Gotzmann's Regularity Theorem is established for globally generated coherent sheaves on projective space. This is used to extend Gotzmann's explicit construction to the Quot scheme. The Gotzmann representation is applied to…

We consider the question of defining interleaving metrics on generalized persistence modules over arbitrary preordered sets. Our constructions are functorial, which implies a form of stability for these metrics. We describe a large class of…

Finitely generated modules over the polynomial ring in $n$ indeterminates are isomorphic to quotients of finite rank free modules. We introduce a theory of relative Gr\"obner bases for those quotients of free modules and, equivalently, for…

We give a self-contained treatment of the theory of persistence modules indexed over the real line. We give new proofs of the standard results. Persistence diagrams are constructed using measure theory. Linear algebra lemmas are simplified…

We prove an index theorem for the quotient module of a monomial ideal. We obtain this result by resolving the monomial ideal by a sequence of Bergman space like essentially normal Hilbert modules.

We show that a persistence module (for a totally ordered indexing set) consisting of finite-dimensional vector spaces is a direct sum of interval modules. The result extends to persistence modules with the descending chain condition on…

We develop some aspects of the homological algebra of persistence modules, in both the one-parameter and multi-parameter settings, considered as either sheaves or graded modules. The two theories are different. We consider the graded module…

We introduce a concept that we call module restriction, which generalizes the classical Weil restriction. We first establish some fundamental properties, as existence and \'etaleness. Then we apply our results to show that Grothendiecks…

We study the decompositions of Hilbert schemes induced by the Schubert cell decomposition of the Grassmannian variety and show that Hilbert schemes admit a stratification into locally closed subschemes along which the generic initial ideals…

Persistence modules serve as the algebraic foundation for topological data analysis, typically studied as representations of posets over a field. This article extends the structural and decomposition theory of persistence modules to the…

A multiplication on persistence diagrams is introduced by means of Schubert calculus. The key observation behind this multiplication comes from the fact that the representation space of persistence modules has the structure of the Schubert…