Betti Functions: An Overview

• Betti functions are assignments that map multi-indices to corresponding homological invariants in graded modules, topological spaces, and combinatorial models.
• They utilize vector-partition functions and chamber decompositions to reveal the quasi-polynomial structure of invariants across rational polyhedral regions.
• They are applied in persistence modules, stochastic topology, and computational algebra to quantify phase transitions and enable stability analyses of topological features.

A Betti function is an assignment associating to each relevant multi-index (typically a degree or threshold parameter) the corresponding Betti number of a family of objects: graded modules, topological spaces, random complexes, excursion sets, or other combinatorial/topological models. Betti functions systematize the dependence of homological invariants on auxiliary data, and encode the structure of minimal resolutions or persistent topological features across parameter ranges. In algebraic contexts, Betti functions track graded or multigraded pieces; in applied topology or random geometry, they characterize the emergence and evolution of topological features as a parameter (e.g. threshold, volume fraction) is varied.

1. Algebraic Betti Functions: Definition and Structural Theorems

In the context of commutative algebra and algebraic geometry, let $S = k[x_1,\dots,x_n]$ be a polynomial ring over a field $k$ with a positive $\mathbb{Z}^d$-grading, and $I \subset S$ a homogeneous ideal with generators $f_1, \ldots, f_r$. For each $t \ge 0$, the $t$-th power $I^t$ is again a graded module. The (multi)graded $i$-th Betti number of $I^t$ is defined by

$$\beta_{i,\mu}(I^t) = \dim_k \operatorname{Tor}^S_i(I^t, k)_\mu,$$

where the subscript $\mu$ denotes the graded piece in degree $\mu \in \mathbb{Z}^d$. The assignment $(\mu, t) \mapsto \beta_{i,\mu}(I^t)$ defines the $i$-th Betti function of the family $\{I^t\}_{t\in\mathbb{N}}$, i.e.,

$$\beta_i : \mathbb{Z}^d \times \mathbb{N} \to \mathbb{N}.$$

The explicit structure of these functions is captured in Bagheri–Lamei's main theorems: for a single ideal and positive $\mathbb{Z}$-grading, there exist finitely many polynomials and a decomposition of $\mathbb{Z}^2$ into regions such that $\beta_{i,p}(I^t)$ is given by these polynomials on each region. For multiple ideals (multigraded case), $\mathbb{Z}^d \times \mathbb{N}^s$ is cut into a finite set of rational polyhedral cones, with $\beta_{i,\mu}(M I_1^{t_1}\cdots I_s^{t_s})$ given, on each cone and appropriate residue classes, by explicit quasi-polynomials (Bagheri et al., 2013).

2. Betti Functions via Vector-Partition Functions and Chamber Decomposition

The combinatorial machinery underlying algebraic Betti functions relies on the theory of vector-partition functions. For a matrix $A$ encoding degrees of generators, the associated function

$p_A(u) = |\{ x \in \mathbb{N}^n : A x = u \}|$

counts lattice points in polyhedral regions, and appears as the coefficient of $t^u$ in $\prod_{j=1}^n (1-t^{a_j})^{-1}$. The cone generated by the columns of $A$ admits a decomposition into finitely many open rational polyhedral chambers $C_1, \ldots, C_r$, on each of which $p_A$ is quasi-polynomial of degree $n-d$. Through the Rees algebra and its Hilbert series, Betti functions are realized as evaluations of vector-partition functions, and thus inherit piecewise quasi-polynomial structure. Each region in parameter space is specified by linear inequalities, and on each, the Betti function is presented by an explicit polynomial or a binomial/Ehrhart-type expansion (Bagheri et al., 2013).

3. Multigraded Betti Functions and Combinatorial/Homological Techniques

For a finitely generated multigraded $R$-module $L$ over $k[x_1,\dots,x_m]$, the $i$-th multigraded Betti function

$$\beta_i : \mathbb{Z}^m \to \mathbb{N}, \qquad \alpha \mapsto \beta_{i,\alpha}(L)$$

records the number of generators in each multidegree in a minimal free resolution. Charalambous–Tchernev describe, under a "generic type" assumption, a combinatorial formula: for generic multidegrees $\alpha$, all but one Betti number in the minimal resolution vanish, and the unique nontrivial Betti number is the $\beta$-invariant of a matroid minor defined by the module structure and the relevant multidegrees. Explicitly, the value $\beta_{i,\alpha}(L)$ equals the dimension of the reduced homology of a certain simplicial complex associated to $L$ and $\alpha$ (a Hochster-type formula), with the homological degree determined by the rank and size of certain subsets of generators (Charalambous et al., 2010).

4. Betti Functions in Persistence Modules, Stability, and Applied Topology

In multiparameter persistence modules (functors $M : \mathbb{R}^n \to \mathrm{Vec}_k$), Betti functions $\beta_{i}: \mathbb{Z}^n \to \mathbb{N}$ capture the structure of minimal free resolutions and encode persistent topological features. Oudot–Scoccola establish that Betti functions are completely determined by the Hilbert function, and are stable with respect to the interleaving distance up to explicit bounds, using a notion of signed-barcode dissimilarity. For $n$-parameter modules, the signed-bottleneck dissimilarity of Betti functions is bounded above by $(n^2-1)$ times the interleaving distance. In the $n=2$ case, a $1$-Wasserstein stability result relates Hilbert functions and Betti barcodes to the $1$-presentation distance (Oudot et al., 2021).

5. Betti Functions for Random Fields and Stochastic Topology

In stochastic geometry and cosmology, Betti functions specify the evolution of topological features within excursion sets of random fields, random simplicial complexes, or models such as the Boolean model or the random connection model. In three-dimensional smooth Gaussian random fields, the three Betti functions $\beta_0(\nu)$, $\beta_1(\nu)$, and $\beta_2(\nu)$ (as functions of threshold $\nu$) measure the number of connected regions, tunnels, and cavities, respectively. Their curves reveal distinct topological phases and transitions—clusters (high $\nu$, $\beta_0$ dominates), tunnels (intermediate $\nu$, $\beta_1$ dominates), and voids (low $\nu$, $\beta_2$ dominates)—and encode more information than the genus alone (Park et al., 2013).

Betti function CLTs for random complexes have been established in the marked random connection model and Boolean model. For a cube $W \subset \mathbb{R}^d$, the appropriately normalized difference of the $p$-th Betti number and its expectation converges to a Gaussian, with explicit mean and variance given in terms of inclusion-exclusion on simplex intersections over the Poisson process (Pabst, 16 Jun 2025). This framework encompasses many functionals arising in random complex models.

6. Betti Functions in Topological Data Analysis and Physical Cosmology

In empirical fields, Betti functions are deployed to systematically distinguish topological phases and transitions. In cosmic reionization, the Betti functions $\beta_0(x)$, $\beta_1(x)$, and $\beta_2(x)$, measured as functions of ionized volume $x$, each peak at physically interpretable stages: growth/merging of isolated regions ($\beta_0$), percolation transition ($\beta_1$), and isolation of void-like neutral regions ($\beta_2$). Analytic forms—log-normal and Gaussian fits—capture model fingerprints and are robust descriptors that outperform conventional metrics like the Euler characteristic, especially when extracted from simulated or observational 21-cm data cubes (Giri et al., 2020).

In large-scale structure analysis, Betti functions deliver finer characterization of topology than the genus curve; changes in their amplitude and shape are sensitive to field statistics and model parameters, enabling detection of non-Gaussianity and topological phase transitions in cosmological data (Park et al., 2013).

7. Betti Functions in Configuration Spaces and Algebraic Geometry

Generating functions for Betti numbers of families of spaces—such as unordered configuration spaces on punctured elliptic curves—take closed rational form, exposing deep analogies with arithmetic zeta functions. For example, $b_i(\mathrm{Conf}^n(E^\times))$ appears as the coefficient of $t^i u^n$ in a rational function $R(t, u)$, mirroring the point-count zeta function of the same space over finite fields. By Deligne purity and spectral sequence analysis, explicit generating functions for Betti and Hodge numbers can be constructed, and purity theorems precisely determine the weight stratification of the cohomology (Cheong et al., 2020).

8. Computational and Algorithmic Aspects

Betti functions offer computational leverage: in the context of powers of ideals, the chamber decomposition and quasi-polynomial structure allow for rapid prediction and computation of Betti tables in high powers, circumventing expensive resolution constructions (Bagheri et al., 2013). In computational algebra, algorithmic frameworks exist for maximizing total Betti numbers under Hilbert function constraints: dynamic-programming (lexsegment ideals, combinatorics of Hilbert difference sequences) produces sharp upper bounds, with package implementations such as MaxBettiNumbers in Macaulay2 (White, 2020).

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References:

• Algebraic structural theory of Betti functions in powers of ideals: (Bagheri et al., 2013)
• Combinatorial formula for graded Betti functions in multigraded modules: (Charalambous et al., 2010)
• Stability of multigraded Betti functions and barcodes: (Oudot et al., 2021)
• Betti functions in random fields and cosmology: (Park et al., 2013, Giri et al., 2020)
• Generating functions and configuration space Betti numbers: (Cheong et al., 2020)
• CLT and variance formulae for Betti functions in random complexes: (Pabst, 16 Jun 2025)
• Algorithmic bounds for Betti numbers via Hilbert functions: (White, 2020)