THE SHIFT-DIMENSION OF MULTIPERSISTENCE MODULES

. We present the shift-dimension of multipersistence modules and investigate its algebraic properties. This gives rise to a new invariant of multigraded modules over the multivariate polynomial ring arising from the hierarchical stabilization of the zeroth total multigraded Betti number. We give a fast algorithm for the computation of the shift-dimension of interval modules in the bivariate case. We construct multipersistence contours that are parameterized by multivariate functions and hence provide a large class of feature maps for machine learning tasks. Acknowledgments. We are grateful to Mats Boij for insightful discussions. W.C. was partially supported by VR, the Wallenberg AI, Autonomous System and Software Program (WASP) funded by Knut and Alice Wallenberg Foundation, and MultipleMS funded by the European Union under the Horizon 2020 program, grant agreement 733161, and dBRAIN collaborative project at digital futures at KTH. R.C. got supported by the Brummer & Partners MathDataLab. A.-L.S. thanks the Brummer & Partners MathDataLab for its hospitality during her research stay at KTH Stockholm in 2021.


Introduction
Topological data analysis (TDA) reveals structures of data by topological methods.One main tool is persistent homology.To the data, one associates a filtered simplicial complex.For every natural number n, one then takes the n -th homology with coefficients in a field K; the ( n -th) persistence module of the data.The persistent homology of a one-parameter filtration is algebraically well understood; by a basic structure theorem from algebra, the persistence module is uniquely determined by its barcode [18] in the discrete setting [14,30] as well as in the real setting [3] under mild finiteness assumptions.Furthermore, the barcode representation of a persistence module is stable [12].From the barcode, one reads the birth and death times of topological features.The case of several parameters, i.e., the study of multifiltered simplicial complexes and their homology as introduced by Carlsson and Zomorodian in [8] allows the extraction of finer information from the data.As was shown in [11], the homology modules of a multifiltered simplicial complex can be obtained as the homology of a chain complex F 1 → F 2 → F 3 of free multigraded modules of finite rank by combinatorial techniques.This allows analysis of multipersistence modules using standard computer algebra software.However, the multiparameter case is algebraically intricate.In contrast to the case of a single parameter, as pointed out in [8], the respective moduli space is not zero-dimensional.Moreover, constructing stable and algorithmic invariants of multipersistence modules is challenging and is currently an active branch of research in topological data analysis.
We investigate a stable invariant of multipersistence modules that is based on the hierarchical stabilization of discrete invariants as introduced by Scolamiero et al. in [27] and by Gäfvert and Chachólski in [17].This construction requires the choice of a distance between multipersistence modules, and it turns a discrete invariant into a measurable real-valued function.The obtained invariant is continuous with respect to the chosen distance on multipersistence modules and with respect to the L p distance on measurable real-valued functions.In this article, we focus on the invariant given by the zeroth total multigraded Betti number β 0 , often called rank of the multipersistence module in a TDA setup, defined as the minimal number of generators.Its hierarchical stabilization is commonly referred to as "stable rank" and was introduced in [17].
For the hierarchical stabilization, we also need to be able to construct distances on the space of tame persistence modules.A way to construct a big class of metrics other than the well-known interleaving distance [23] and matching distance [2,21] is via so-called persistence contours.Persistence contours are rather well-studied in the one-parameter case.In [10], for example, the authors list various contours and show how classification, using the first one-persistence homology group, of point processes in the unit square (sampled with respect to different probability distributions) can be improved by choosing the right contour.For the case of several parameters, only a small number of examples of persistence contours is known besides the standard contours.The latter are parameterized by vectors.Persistence contours can be described by simple noise systems [17,27].Those are closely related to so-called "superlinear families" that were introduced independently in [6], a generalization of which can be found in [4].Therein, no extensive class of parameterized such objects was given yet, but it is essential for machine learning tasks to have such a class at one's disposal.
Multigraded modules over the polynomial ring K[x 1 , . . ., x r ] with the standard multigrading can be considered naturally as multipersistence modules.In this article, we investigate the hierarchical stabilization of β 0 for multipersistence modules obtained in this way with respect to a broader family of contours.The obtained invariant is called shift-dimension.
The main contribution of this article is twofold.The first main contribution is the realization that the shift-dimension can be constructed and defined entirely algebraically without referring to the geometry of multipersistence modules induced by the choice of a distance.We give an explicit description of the shift-dimension and investigate some of its algebraic properties.We thereby establish a connection between topological data analysis and combinatorial commutative algebra.In general, the computation of the stable rank is algorithmic but NP-hard [17].For the bivariate case, we present a linear-time algorithm for the computation of the shift-dimension of interval modules and therefore for the computation of the shift-dimension of quotients of monomial ideals.
The second main contribution lies in the construction of multipersistence contours.We present a class of contours that is parameterized not only by vectors, but by multivariate functions.In doing so, we provide a large class of multipersistence contours.
Related work.The investigation of invariants of multipersistence modules is currently a prominent topic in TDA.For two parameters, the Hilbert function and the rank invariant of [8] are implemented in RIVET [28] for an interactive data analysis.It has been used in applications such as [20].Furthermore, minimal presentations can be computed efficiently for big data sets using RIVET [24] or mpfree [22].Invariants such as the Hilbert series, associated primes, and local cohomology have been investigated [19], as well as the multirank function [29] and the compressed multiplicity [15].
Outline.This article is organized as follows.In Section 2, we review basic notions about multipersistence modules, how to measure distances between them, and the hierarchical stabilization of discrete invariants.We present several new contours for the multiparameter case.Section 3 introduces and studies the shift-dimension of tame persistence modules from an algebraic and combinatorial perspective.In particular, we focus on the investigation of (non-)additivity.We present a linear-time algorithm for the computation of the shift-dimension of interval modules in the bivariate case.In Section 4, we study the corresponding invariant of multigraded modules over the multivariate polynomial ring and investigate its algebraic properties.In Section 5, we give an outlook to open problems and future work.
Let K[G] denote the monoid ring of G over the field K with its natural G -grading.The category Mod gr (K[G]) of G -graded K[G] -modules is isomorphic to the thin category Fun (G, Vect K ) of functors from the posetal category G to the category of K -vector spaces Vect K as follows.To each M = ⊕ g∈G M g ∈ Ob(Mod gr (K[G])), one associates the functor (g → M g ) ∈ Ob(Fun (G, Vect K )).Each graded morphism φ = (φ g ) g∈G : naturally gives rise to a natural transformation {φ g : M g → N g } g∈G of the corresponding functors.The objects of those categories are called (multi-)persistence modules.We are going to switch seamlessly between both points of view.We denote by Tame (G, Vect K ) the subcategory of Fun (G, Vect K ) which corresponds to the subcategory Mod gr f.p. (K[G]) of Mod gr (K[G]) consisting of finitely presented such modules, as investigated in [13].In this article, we often consider interval modules which are defined as those elements of Fun(G, Vect K ) that correspond to quotients of two monomial ideals.A subset I ⊆ G is called interval in the poset (G, ≤) if for any g, h ∈ I and any f ∈ G the following two conditions hold: A free persistence module is a module of the form F = ⊕ g∈G K(g, •) β 0 (g) , where β 0 (g) ∈ N and K(g, •) ∈ Fun(G, Vect K ) denotes the functor The natural number β 0 (F ) := g∈G β 0 (g) is called the rank of the free module F.
Definition 2.1.Let M ∈ Fun((G, ≤), Vect K ) be a persistence module.The total zeroth multigraded Betti number of M, denoted β 0 (M ) is the smallest possible rank of a free persistence module F such that there exists a surjection from F onto M.
Regarding M as an R r -graded K[R r ] -module, the number β 0 (M ) is the minimal number of homogeneous generators of M. Caveat 2.2.In TDA literature, one often refers to β 0 as "rank".This is not consistent with standard terminology in algebra.Yet, they do coincide for free modules.
2.2.Distances via persistence contours.It is crucial to be able to measure distances between persistence modules.In order to learn metrics in the sense of machine learning, one needs a large space of such metrics parameterized by simply describable objects.In this section, we recall a method to construct metrics on the category of tame functors, namely in terms of so-called persistence contours.Denote by R the poset R ≥0 and by R r ∞ , where r ∈ N, the poset obtained from R r by adding a single element ∞ such that x ≤ ∞ for all x ∈ R r and setting ∞ and ε, τ ∈ R, the following two conditions hold: The second property in the definition indicates that a persistence contour fulfills a lax action only, i.e., the inequality is in general not an equality.For brevity, one often refers to persistence contours as just "contours".One common example is the standard contour in the direction of a vector v ∈ R r defined as C(x, ε) := x + εv; i.e., one shifts x by the vector v scaled by ε.An immediate generalization of that contour is to travel along a curve instead of a line as explained in Example 2.9.
Following an immediate generalization of [10, Section 3], we now recall how persistence contours give rise to pseudometrics on the category of persistence modules.Definition 2.4.Let C be a persistence contour, M, N ∈ Tame(R r , Vect K ) tame persistence modules, and ε ∈ [0, ∞).A morphism f : M → N is an ε -equivalence if for every a ∈ [0, ∞) r s.t.C(a, ε) ̸ = ∞, there is a K -linear function N (a) → M (C(a, ε)) making the following diagram commutative: Two tame functors M and N are called ε -equivalent (with respect to C ) if there is a tame functor L and morphisms f : M → L ← N : g such that f is an ε 1 -equivalence, g is an ε 2 -equivalence, and Denote by E := {ε ∈ [0, ∞) | M and N are ε-equivalent}.Then the function is an extended pseudometric on Tame(R r , Vect K ).There is an alternative way of thinking of contours.For that, we now recall the definition of a noise system from [27].Definition 2.6.A noise system in Tame(R r , Vect K ) is a collection {V ε } ε∈R of sets of tame functors such that the following three conditions are satisfied: (i) For any ε, the zero functor is contained in V ε , (ii) for all 0 ≤ τ < ε, V τ is contained in V ε , and Throughout this article, we will mainly focus on noise systems {V ε } that are closed under direct sums, i.e., noise systems such that whenever M, N ∈ V ε , also their direct sum If {V ε } is closed under direct sums, a minimal element in B(M, τ ) is unique if it exists; this element is denoted by M [τ ] and is called the τ -shift of M. Then B(M, τ ) coincides with the set Under this correspondence, for the standard contour in the direction of a vector v, M [τ • v] is (τ • v) * M, where * denotes the induced action of the monoid R r on the monoid ring K[R r ] .

Construction of new contours.
In the setting of a single parameter, there are various contours suitable for concrete data-analytic tasks [10].In this subsection, we construct several contours for the multiparameter case.
Example 2.9 (Curve contour).Consider a curve in R r that is given by a monotone non-decreasing function I : R → R r for which I(0) lies in one of the coordinate axes.Assume further that R r can be covered by translations of that curve parallel to r − 1 coordinate axes such that every point in R r lies on precisely one of the curves.For x ∈ R r ∞ and ε ∈ [0, ∞), define C(x, ε) to be the point that one obtains by traveling along the translated curve that runs ε -far through x ; i.e., if I x denotes the translated curve running through x at ε 0 , we set C(x, ε) to be I x (ε 0 + ε).♢ For some further examples of persistence contours in the one-parameter case, we refer to [10].Among them is the contour of distance type.Fixing a vector, we can immediately generalize this contour to the multiparameter setting.
Example 2.10 (Distance type in a fixed direction).Let r ∈ N, x, v ∈ R r , and f a non-negative (Lebesgue-)measurable function on R r .For denotes the set to which we refer to as L-shape (at x ).♢ Remark 2.11.Note that the L-shapes in Example 2.10 can not be replaced by the rectangles of the same bounds; it would not define a lax action anymore.♢ Applying a one-parameter contour component-wise, as in the following example, yields a multiparameter contour.
Example 2.12 (Component-wise shift type).Let r ∈ N, f 1 , . . ., f r : [0, ∞) → [0, ∞) be non-negative measurable functions and x ∈ R r .For each x i , set y i to be the infimum of all ỹi ∈ R such that x i = ỹi 0 f i dL 1 , if such ỹi exists.Let m 1 , . . ., m r be super-additive non-negative functions.One obtains a contour by setting We now introduce a generalization of the contour of shift type that is based on multivariate functions.This yields a large class of persistence contours.

Example 2.13 (Multivariate shift type
, where inf denotes the meet in the poset.Let v ∈ R r .One obtains a contour by setting where sup denotes the join of the poset.If B(x) is empty, set C(x, ε) := ∞.To verify that C is a contour, one can observe that, for every x and ε , the element b(x) for every i .This inequality implies that condition (ii) of Definition 2.3 is fulfilled.
The computational bottleneck is to compute b(x).This, however, can be done efficiently by an approximation scheme to arbitrary precision by studying the intersection of B(x) with an equidistant grid and successively decreasing the grid length.♢ 2.4.Hierarchical stabilization.In this section, we recall from [10,17] the concept and properties of the hierarchical stabilization.This process turns discrete into stable invariants as follows.Let T be a set and d : T × T → R ∪ ∞ an extended pseudometric on T, where again R denotes the poset of non-negative real numbers.Let f : T → N be a discrete invariant and denote by M the set of Lebesgue-measurable functions from [0, ∞) to [0, ∞), endowed with the interleaving distance as in [17, Section 3].
Definition 2.14.Fix an extended pseudometric d on T. The hierarchical stabilization of f , denoted f , is defined to be f : Hence, f (x) ∈ M is the monotone, non-increasing measurable function that maps a real number τ to the minimal value that the discrete invariant f takes on a closed τ -neighborhood of x.Note that f strongly depends on the chosen pseudometric on T.
Proposition 2.15.[10, Proposition 2.2] For any choice of extended pseudometric on T, the function f : Therefore, the hierarchical stabilization turns discrete into stable invariants.

The shift-dimension of multipersistence modules
In this section, we introduce the shift-dimension of persistence modules and investigate its algebraic properties.We particularly focus on (non-)additivity.
A v -basis of M is a set of v -generators m 1 , . . ., m k such that k is smallest possible.In this case, we call k the shift-dimension (or v -dimension) of M and denote it dim v (M ).
Throughout the article, we are going to stick to homogeneous elements when studying generators and v -bases.The following lemma justifies that convention.Lemma 3.2.If M is a tame persistence module, then for all v ∈ R r there exists a v -basis consisting of finitely many homogeneous generators.
Proof.This statement immediately follows from the fact that every finitely presented R r -graded K[R r ] -module can be minimally generated by homogeneous elements.□ The shift-dimension can be equally characterized as the smallest number of elements of M such that v * M is contained in the submodule generated by those elements.
Remark 3.3.The 0 -dimension of M is the minimal number of generators of M. For a fixed vector v, one may consider the family {dim τ v (M )} τ ∈R ≥0 of shift-dimensions of a multipersistence module M. The non-increasing function R ≥0 → N mapping τ → dim τ v (M ) is suitable as a feature map for machine learning algorithms.♢ In general, R r -graded K[R r ] -modules are hard to handle.We will mostly restrict to the category of finitely presented R r -graded K[R r ] -modules and thereby to the category Tame (R r , Vect K ) of tame functors.
Example 3.4 (An indecomposable).Consider the following commutative diagram of vector spaces and linear maps: This example is taken from [7] and slightly modified.Maps between identical vector spaces are defined to be the identity map.We extend this representation to a two-parameter persistence module M by reading the vertices of the quiver among {(i, j)} i=0,...,11, j=0,...,6 as depicted above, by defining M (i,j) to be the trivial vector space for all remaining points of N 2 , and to have identity maps id : (i, j) → (i+ε 1 , j+ε 2 ) for all i, j ∈ N, ε 1 , ε 2 ∈ [0, 1).This persistence module is indecomposable, i.e., it is not the direct sum of any two non-trivial persistence modules.Let v = (2, 1).Then dim 0 (M ) = β 0 (M ) = 5, dim v (M ) = 2, and a v -basis is given by generators of the vector spaces at degrees (0, 4) and (6,1).♢ At first glance, the notion of the shift-dimension might seem rather artificial.But, in fact, it arises in a natural way; namely as the hierarchical stabilization of β 0 .Proposition 3.5.Let M ∈ Tame(R r , Vect K ) be a tame persistence module.Denote by β 0w the hierarchical stabilization of the zeroth total multigraded Betti number w.r.t. the standard noise in the direction of 0 ̸ = w ∈ R r and v = w ∥w∥ its normalization.Then Proof.Denote by C the standard contour in the direction of v.As in [27, Section 6.2], we denote by the noise system associated to the standard contour C. The system {V δ } δ∈R is called the standard noise in the direction of v .Then, by [17,Theorem 8.3], Addition of the vector δv in the definition of V δ corresponds to the action of δv on the module.Hence, in our case, Furthermore, in [17,27], noise systems are described for the non-negative rational numbers instead of the non-negative real numbers.The corresponding arguments we use generalize to Tame(R r , Vect K ).♢ Remark 3.7.As shown in [17,Section 11], computing the shift-dimension can be reduced to an NP-complete problem and hence is NP-hard itself: for a certain class of persistence modules, the computation of the shift-dimension is at least as hard as the RANK-3 problem [26], an optimization problem in linear algebra.♢ We obtain a truncated version of the shift-dimension as stabilization of the zeroth multigraded Betti number by changing the contour to be truncated.For C a contour and α ∈ R r , denote by C α the truncation of C at α.This is defined as We call the corresponding noise system the α -truncated noise system.
Proposition 3.8.Under the assumptions of Proposition 3.5 with α -truncated noise in the direction of a normalized vector v, β 0 stabilizes as follows with respect to the α -truncated standard contour in the direction of v : Proof.The noise system associated to the truncated standard contour is given by We denote the balls corresponding to this noise system by B α .Applying [17, Theorem and thus the claim of the proposition follows.□ Therefore, computing the stabilized β 0 with respect to a truncated contour corresponds to truncating the module at degree α.

3.2.
Properties of the shift-dimension.We now state some fundamental properties of the shift-dimension.In particular, we investigate it regarding additivity.
On the level of epimorphisms, the following holds true.
Lemma 3.9.Let φ : M ↠ N be an epimorphism of tame persistence modules.Then the following holds true: For monomorphisms, we do not get a corresponding inequality in the reverse direction, as the following counterexample demonstrates.
Example 3.10.Let v = (1, 1), M be the interval module generated in degrees (0, 2) and (2, 0), and N the free module generated by a single element in degree (0, 0).Then Proof.Choose a w -basis {n 1 , . . ., n a } of N. For every i, choose l i ∈ L for which ψ(l i ) = n i .Let {g 1 , . . ., g β 0 (L) } generate L. Then for all i there exist α 1 , . . ., α a ∈ K[R r ] such that w * g i − a j=1 α j l j ∈ ker(ψ) = im(φ).Choose a v -basis {m 1 , . . ., m b } of M. Thus, for all i, there exist γ 1 , . . ., For the proof of statement (ii), let F be the free module generated by a v -basis of N. Since ψ : L ↠ N is surjective, we obtain a morphism F → L and hence a morphism from In general, it is not additive, as the following counterexample for L-shapes demonstrates.

Example 3.14 ( r = 2 ). Consider the interval modules
♢ We now present some cases for which additivity of the shift-dimension does hold true.
Lemma 3.15.If F and G are two free persistence modules of finite rank, then Proof.The statement follows from the fact that in sufficiently large degree, the elements of the v -basis need to generate K β 0 (F )+β 0 (G) .□ Proposition 3.16.Let M be an interval module generated by one element and F a free persistence module of rank one.Then Proof.For dim v (M ) = 0, the statement is clear.Now assume that dim v (M ) = 1.
Assume there exists a v -basis {(m 1 , f 1 ), . . ., (m k , f k )} M of cardinality k.Since dim v (F ) = k and F is free, f 1 , . . ., f k need to be linearly independent.By assumption on the shift-dimension of M, there exists m ∈ M s.t.v * m ̸ = 0.Because of the linear independence of f 1 , . . ., f k , the element (v * m, 0) is not contained in the submodule generated by (m 1 , f 1 ), . . ., (m k , f k ), in contradiction to the assumption.□

3.3.
A quantitative study of non-additivity.In this subsection, we investigate non-additivity of the shift-dimension in greater detail and give a measure for it.For a finite family of persistence modules {M i } i∈I and a fixed, possibly learned v ∈ R r , we define the locus of non-additivity of the shift-dimension as For a quantitative study, we associate to {M i } i∈I the L p -distance between the functions i∈I dim • (M i ) and dim • i∈I M i of τ, i.e., for 1 ≤ p < ∞, we study If this expression is sufficiently small, the sum of the shift-dimensions of the M i yields a good approximation of the shift-dimension of their direct sum ⊕M i .Revisiting and generalizing Example 3.10, we now undertake more quantitative investigations.
Example 3.18.Let v ∈ R r >0 .Let M 1 , M 2 ̸ = 0 be interval modules generated by single elements g 1 and g 2 , resp., and quotiented out by single elements g 1 and g 2 , resp.Assume further that lcm(deg Let M 1 , M 2 be interval modules as in the example above.In this case, the union of their underlying intervals is an interval.Denote by M the corresponding interval module.Then, whenever dim The following example demonstrates that the difference between dim v (⊕ i∈I M i ) and i∈I dim v (M i ) can be arbitrarily large.In fact, the shift-dimension of a module can be 1 while the Hilbert function and the rank invariant attain arbitrarily large values.Left: Let v = (4, 4).All shifted minimal generators have pairwise incomparable degrees but can be generated by the all-one vector in the degree of the greatest common divisor of their degrees.Hence, dim v (⊕ i (M i )) = 1 .Right: Example 3.19.Let M 1 and M 2 be as in Example 3.18.Define a finite collection of interval modules {M i } i∈I such that each M i has one generator g i and gets quotiented out by two elements g i , g i that are of the following form.The {g i } i∈I are assumed to have pairwise distinct degrees which lie on the straight line between deg(g 1 ) and deg(g 2 ).The degrees of g i and g i are chosen such that deg( Then, again, the all-one vector in degree lcm(deg(g 1 ), deg(g 2 )) is a divisor of each v * (0, . . ., g i , . . ., 0).Hence, dim v ( {1,2}∪I M i ) = 1 ̸ = 2 + |I| = {1,2}∪I dim v (M i ).See Figure 1 for an illustration of one concrete example.♢ Note that Loc v ({M i } i∈I ) is in general not connected.Counterexamples can be constructed by taking direct sums of certain modules as in the examples above.
We give a further, more involved example for which additivity does not hold true.
Example 3.20 (Example 3.4 revisited).Let M 1 be the persistence module from Example 3.4 and M 2 be an interval module generated at degrees (0, 5) and (5, 1.5) and quotiented out at degrees (2,6) and (9, 1.5).Let v = (2, 1).Then, We get Loc v ({M 1 , M 2 }) = [1.5, 2) and err v,p ({M 1 , M 2 }) = 0.5 1/p .♢ Here, the non-triviality of the loci and L p -errors of non-additivity is due to the existence of v -basis elements such that a suitable multiple of them yields a non-zero shifted generator in one direct summand and zero on the other direct summands.

3.4.
Basis exchange properties.Unlike bases of vector spaces in linear algebra, v -bases do not give rise to a matroid if r > 1.The following example demonstrates that even for monomial ideals, neither the well-desired basis exchange property holds, nor that for every set ot v -generators there exists a subset which is a v -basis of M.
Example 3.21.Let M be the interval module generated at degrees (0, 8), ( Proof.Let b be as in the assumption of (i).Then (B\{b})∪{ b} is a set of v -generators as it generates ⟨B⟩ by construction.Since it has the same cardinality as B, it is indeed a v -basis.For the proof of the second statement, let w be as desired.We prove that for each g Given b ∈ B, there is-up to constants in the field K -a unique maximal multiple of b by which one can replace it, but there might be several incomparable such choices for divisors of b.If one replaces a v -basis element by a multiple or by a divisor of it as described in the lemma above, it is important to note that the indispensability relations of the other v -basis elements with the minimal generators of M might change.Indispensability therefore encodes the combinatorial complexity of v -bases.
Corollary 3.23.Let M be an interval module, G a minimal set of homogeneous generators of M, and B a v -basis of M. Then the following statements are true.
(i) There exists a v -basis consisting of minimal generators of M.
Proof.Since M is an interval module, each homogeneous element is a multiple of some element of G. Using Lemma 3.22 iteratively, replace each element of B by a respective element of Note that the reverse of the second statement does not hold true.

3.5.
Algorithm for interval modules.The software Topcat [16] of Gäfvert provides tools for calculating stable ranks for arbitrary persistence modules and, in principle, is based on searching through the entire B(M, δ) to look for an element with the smallest β 0 .This search can be done more efficiently for interval modules in the twoparameter case, and we now present its algorithm.An implementation in C++ of our algorithm is made available at https://github.com/recorb/shiftdimstablerank.
, M be an interval module, and G := {g i } i∈I⊆R be the generating curve of M. Denote deg(g i ) = (a i , b i ).W.l.o.g.assume that the generators are listed in the following total order: a i < a j or b i > b j whenever i < j.Algorithm 3.24.
While there exists i ∈ I k such that v * g i ̸ = 0, do: (a) Set i k to be the maximum of the set (c) Replace k by k + 1 and iterate.
Note that the computation of the i k and I k+1 can be carried out geometrically: to obtain i k , we project from down to the generating curve of M. For the construction of I k+1 , we project to the right from deg(g i k ) − v to the generating curve and hit a coordinate (a, b).We remove all generators whose second coordinate is not strictly greater than b from I k to obtain I k+1 .See Figure 3 for an illustration of this geometric procedure.Proof.First note that by Corollary 3.23 (i), there is a subset of G that is a v -basis of M. Hence, a minimal subset H of G such that v * g i ∈ ⟨H⟩ for all i ∈ I is indeed a v -basis.If H = ∅, the statement is clear.Hence, let ℓ > 0. Let us first argue that ℓ is finite.Since v 1 ̸ = 0, i 1 is well-defined.For all 0 We now prove that ℓ = dim v (M ) and H is a v -basis of M. We have v * (M/⟨H⟩) = 0. Hence, dim v (M ) ≤ ℓ.For the reverse direction, assume that ℓ > 0 and let B := {b j } j∈J be a v -basis of M. Applying Lemma 3.22 iteratively, we can w.l.o.g.assume that for each j, b j = g i for some i ∈ I. Hence, w.l.o.g.B is a subset of G that might differ from H.However, B clusters I into dim v (M ) subsets; the cluster labeled by j ∈ J consists of all i ∈ I for which deg(b j ) ≤ (deg v * g i ).If two elements of I both are elements of one such cluster, then each element in between must be in the same cluster as well.By construction, for all 1 ≤ k ≤ ℓ there is no i ∈ I \ ( 1≤j≤k−1 I j ) such that there exists Ĩk ⊋ I k with Ĩk ∩ ( 1≤j≤k−1 I j ) = ∅ and ⟨{v * g j | j ∈ Ĩk }⟩ ⊆ ⟨g i ⟩.Hence, the cluster given by B which contains the smallest elements of I must be a subset of I 1 .By construction of I 2 , the second cluster given by B can not be a proper superset of I 1 ∪ I 2 .Iteratively, we get that the cardinality of B is at least ℓ.□ Replacing the total order on the generators by the opposite total order yields another variant of the algorithm computing dim v (M ).This yields the upper bound ℓ ≤ ⌈ The algorithm can be extended to the case v 1 = 0, v 2 ̸ = 0 if sup{b i | i ∈ I} < ∞ and-using the opposite total order-to the case For the complexity of the algorithm, it remains to investigate the complexity of Steps 1(a) and 1(b).By the geometric construction described above, we need to compute infima, suprema, add and subtract a vector, project to a curve, and check if certain segments of a curve get shifted beyond another curve.In general, computing the shiftdimension is NP-hard, as was mentioned in Remark 3.7.We are now going to prove that the computation is linear for tame interval modules.Proof.Each iteration of Step 1 boils down to carrying out two additions, determining three minima of a finite array of sorted data, and deciding whether v * g j = 0 only for a subset of minimal homogeneous generators.Deciding whether v * g j = 0 can again be done by finding the minimum of a finite array of sorted data.The total number of the latter operations can naively bounded by β 0 (M ) and can hence be done as a preprocessing step of β 0 (M ) operations.All the individual operations can be done in O(1) time.Running the algorithm from above or below, the number of iterations of Step 1 is bounded by ⌈ ⌉, respectively.Since β 0 (M ) is finite, it is an additional upper bound for the number of iterations of Step 1. □ For some interval modules, such as monomial ideals, the module generated by H has an illustrative description: it is a staircase with the minimal possible number of steps that fits between M and v * M.This investigation might give additional ideas for a proof of an analogous algorithm in the setting of more than two parameters.Such a generalization would presumably increase the geometric complexity of the algorithm.
The insights of this algorithm in combination with a better structural understanding of the examples in Section 3.3 might give rise to an efficient way to compute the shiftdimension of direct sums of interval modules.
4. The shift-dimension of multigraded K[x 1 , . . ., x r ] -modules In this section, we explain how the shift-dimension for tame persistence modules translates to the monoid G = N r .Identifying the monoid ring K[N r ] with the polynomial ring K[x 1 , . . ., x r ] and hence translating v * (•) to multiplication by the monomial ) is tantamount to the study of multigraded modules over the multivariate polynomial ring.Those modules play a fundamental role in algebraic geometry.We denote by S := K[x 1 , . . ., x r ] the polynomial ring in r variables over a field K with the standard N r -grading, i.e., the degree of x i is the i -th standard unit vector e i ∈ N r .Motivated by the stabilization of β 0 with respect to the metric arising from the standard contour in the direction of a vector, we introduce the following invariant of multigraded S -modules.
and for which k is smallest possible.In this case, we call k the v -dimension of M and denote it by dim v (M ).
Again, for finitely generated persistence modules, we stick to homogeneous generators and v -bases.For fixed v ∈ N r , we abbreviate dim nv by dim n and refer to it as n -dimension, and similarly for " n -basis" and " n -generators".Remark 4.2.This definition generalizes to arbitrary rings R. Let M be an R -module and r ∈ R. Elements {m 1 , . . ., m k } of M r -generate M if rM ⊆ ⟨m 1 , . . ., m k ⟩. ♢ Note that dim 0 (M ) is the minimal number of (homogeneous) generators of M.Moreover, dim n (M ) is zero if and only if (x v 1 1 • • • x vr r ) n M is the zero-module.Remark 4.3.By Hilbert's syzygy theorem, every finitely generated N r -graded S -module has a minimal free resolution F • of length at most r (see [25,Proposition 8.18]).The rank of F i is the i -th total multigraded Betti number of the module.It would be intriguing to stabilize higher total multigraded Betti numbers as well.♢ The following proposition explains how to modify tame persistence modules to obtain a module in which every element can be extended to an n -basis.This construction works for the monoid ring K[N r ], but not for K[R r ].We now present some cases for which additivity of the shift-dimension does hold true.They all are immediate consequences of statements presented in Section 3.2.Here, we rephrase the statements for the discretized setting and omit the proofs.

Outlook to future work
Our article outlines several pathways for future research.On the one hand, it would be intriguing to stabilize discrete invariants other than the zeroth total multigraded Betti number, such as the rank of the highest syzygy module.The following question immediately arises: Is the rank of the highest syzygy module "naturally" linked to a noise system-similar to the zeroth total multigraded Betti number being linked to the noise system in the direction of a vector in the sense of Definition 2.7?A further challenging question is whether the stabilized Euler characteristic is the alternating sum of the stabilization of the ranks.
On the other hand, it would be worthwhile to investigate the stabilization of β 0 with respect to pseudometrics other than the one arising from the standard contour.For that, a first step is the description of the shift of the module.As described, for the standard contour in the direction of a vector, this is the module multiplied by the corresponding monomial.For other contours, there is no such algebraic description yet.
Another interesting direction would be to investigate the shift-dimension for (not necessarily graded) modules over arbitrary rings and to develop a geometric intuition.
We believe that the shift-dimension allows for the extraction of finer and new information of data.This makes it valuable for applications in the medical and life sciences, among others.In order to compute the shift-dimension of multipersistence modules of actual data arising in the sciences efficiently, one needs to extend the linear-time algorithm presented in this article to modules other than interval modules, such as direct sums of interval modules.Those indeed arise from data [15] and approximate persistence modules in the sense of [1], [5], and [9].Using our algorithm, evaluating the shift-dimension summand-wise is already possible for this class of modules.It would be worthwhile to investigate if this additive version is stable, as was suggested to us by Ulrich Bauer.Furthermore, it would be interesting to find bounds for err v,p and to determine subclasses of modules for which this error is zero.
In the one-parameter case, two persistence modules M and N are known to be isomorphic if and only if β 0C (M ) and β 0C (N ) coincide for each persistence contour C. At present, it is not known if that statement holds true for the multiparameter case as well.As a first step in this direction, one might investigate whether one can distinguish non-isomorphic multigraded modules in the following sense: given two non-isomorphic modules M, N, can one find a noise system (V ε ) ε∈R ≥0 such that M ∈ V 0 but N / ∈ V 0 ?

Definition 2 . 7 (
[17, Definition 8.2]).A noise system is called simple if (i) it is closed under direct sums, (ii) for any tame functor M and any τ ∈ R, B(M, τ ) contains the minimal element M [τ ], and (iii)β 0 (M [τ ]) ≤ β 0 (M ) for any τ ∈ R.The third condition inherently links simple noise systems to β 0 .Proposition 2.8 ([17, Theorem 9.6]).For a persistence contour C, denote by V C,ε the set of tame functors M such that M (x ≤ C(x, ε)) is the zero-morphism whenever C(x, ε) ̸ = ∞.The function C → {V C,ε } ε is a bijection between the set of persistence contours and the set of simple noise systems.

8 . 3 ]
, we get β 0v,α (M )(δ) = min {β 0 (U ) | U ∈ B α (M, δ)} .We have M/U ∈ V δ,α iff for all homogeneous m ∈ M/U we have δv * m = 0 or α ≤ deg(δv * m).Hence, taking the quotient by a submodule that is generated by a single element can drop the shift-dimension by one at most.Proof.Assume there exist m 1 , . . ., m dimv(M )−2 ∈ M/⟨m⟩ that are a v -basis of M/⟨m⟩.Then for any choice of representatives mi of m i , m, m1 , . . ., mdimv(M)−2 v -generate M. If follows that dim v (M ) ≤ dim v (M ) − 1, which is a contradiction.□In order to decide whether an element m ∈ M can be extended to a v -basis {m, m 2 , . . ., m dimv(M ) } of M, one makes use of the following criterion.Lemma 3.12.Let v ∈ R r and m ∈ M.There exists a v -basis of M containing m if and only

Figure 1 .
Figure 1.Illustration of a persistence module as in Example 3.19 for |I| = 2 .Left: Let v = (4, 4).All shifted minimal generators have pairwise incomparable degrees but can be generated by the all-one vector in the degree of the greatest common divisor of their degrees.Hence, dim v (⊕ i (M i )) = 1 .Right:The functions i dim τ v (M i ) (in lavender) and dim τ v (⊕ i (M i )) (in blue) for v = (1, 1).We have Loc v ({M i } i∈I = [3, 4.3) and err v,p = (0.6 • 2 p + 0.7 • 3 p ) 1/p .

Corollary 3 . 26 .
If M is a finitely presented interval module in the bivariate case, then dim v (M ) can be computed in O(n) time, where n := β 0 (M ).

Definition 4 . 1 .
Let M be an N r -graded S -module and v

Proposition 4 . 4 .
Let M be a finitely generated N r -graded S -module and n ∈ N. Choose a non-zero element m 1 ∈ M that is not contained in any n -basis of M and consider the quotient module M/⟨m 1 ⟩.Then choose a non-zero element [m 2 ] ∈ M/⟨m 1 ⟩ that is not contained in any n -basis of M/⟨m 1 ⟩, and repeat this process.There exists a natural number ℓ such that after ℓ iterationsM −→ M/⟨m 1 ⟩ −→ M/⟨m 1 , m 2 ⟩ −→ • • • −→ M/⟨m 1 , . . ., m ℓ ⟩ =: M ,
0} , and the claim follows.□ Note that β 0v (M ) is a monotone, non-increasing step function.