Varieties of Polynomial Growth

• Varieties of polynomial growth are defined by functions scaling as O(n^k), encapsulating growth patterns in algebraic, geometric, and combinatorial settings.
• The topic spans applications in algebraic geometry, PI-theory, and group theory, utilizing techniques such as Newton–Okounkov bodies and Bass–Guivarc’h formulas.
• Classification results reveal clear dichotomies between polynomial and exponential regimes, with minimal examples delineating boundaries in structural complexity.

A variety of polynomial growth arises whenever the size, dimension, or another invariant of a mathematical or algebraic structure grows as a polynomial function of a natural parameter. Such growth rates appear in fields including algebraic geometry, group theory, ring theory, PI theory, and arithmetic geometry. Precisely, a sequence $\{a_n\}$ is said to exhibit polynomial growth of degree $k$ if $a_n = O(n^k)$ as $n \to \infty$, possibly with explicit leading coefficients. The notion underpins modern classification theorems for algebras, varieties, and groups, and reveals rich structural dichotomies between polynomial, subexponential, and exponential regimes.

1. Algebraic and Geometric Frameworks for Polynomial Growth

Varieties of polynomial growth prominently feature in the asymptotic analysis of graded structures on algebraic varieties and algebras. In algebraic geometry, the section ring $$R(\mathcal{L}) = \bigoplus_{n\geq 0} H^0(X, \mathcal{L}^n)$$ of a line bundle $\mathcal{L}$ on a $d$-dimensional projective scheme $X$ over a perfect field $k$ provides a prototypical instance. For a graded linear series $L = \bigoplus_{n\geq 0} L_n \subset R(\mathcal{L})$, the dimension function $h^0(X, L_n)$ encodes the growth, and the Kodaira–Iitaka dimension $q(L)$ governs the leading term. Classical results guarantee that for reduced $X$, $h^0(X, L_n) = \Theta(n^q)$ for $q = q(L)$, with the leading coefficient determined by intersection theory or Newton–Okounkov convex bodies (Cutkosky, 2012).

In dynamical algebraic geometry, polynomial growth rates quantify the complexity of morphisms. For a surjective self-morphism $f: X \to X$ on a smooth projective $d$-fold, the $k$-th dynamical degree $\lambda_k(f)$ and the associated algebraic entropy $h_{\rm alg}(f)$ provide invariants, with zero entropy if and only if all $\lambda_k(f) = 1$. The polynomial volume growth $\mathrm{plov}(f)$, defined via the volumes of graphs $\Gamma_n$ and sums of divisors $\Delta_n$, measures how the induced volumes scale (as $\deg((\Delta_n)^d)$) and is linked to the Gelfand–Kirillov dimension of certain twisted homogeneous coordinate rings (Hu, 2022). For abelian varieties with quasi-unipotent automorphisms, such growth rates admit explicit block-diagonal formulas in terms of Jordan block sizes in the Tate module action, and the spectrum of possible growth rates is completely classified.

In arithmetic geometry, a projective variety $X \subset \mathbb{P}^n_\mathbb{Q}$ has rational points of height at most $B$ growing as $N_X(B) = O(B^k)$ for some $k$, often $\dim X$. The optimal form, $N_X(B) \le c(n) d^{e(n)} B^{\dim X}$, was established without extraneous $\varepsilon$ or log factors, and lower bounds prove the polynomial exponent in $d$ is necessary (Castryck et al., 2019).

2. Polynomial Growth in Algebraic PI-Theory

In ring theory and PI-theory, varieties of algebras are classified by their codimension growth. For a variety $\mathcal{V}$ defined by a $T$-ideal of polynomial identities, the $n$-th codimension $c_n(\mathcal{V})$ is the dimension of the $n$-variable multilinear polynomials modulo identities. Polynomial growth of degree $t$ means $c_n = O(n^t)$. Exponential growth, by contrast, corresponds to the existence of a limit $\lim_{n \to \infty} c_n^{1/n} > 1$.

Kemer-type theorems and their graded, involutive, and generalized analogues establish a dichotomy: the codimension sequence is either polynomially bounded or grows exponentially, with no intermediate rates (Cota, 5 Dec 2025, Cota et al., 4 Dec 2025, Ioppolo et al., 2020, Martino et al., 18 Feb 2025). The structure theorem often states that polynomial growth occurs if and only if the semisimple part is at most one-dimensional in each direct summand, and the radical controls the degree: if the nilpotency index is $q+1$, the degree is $q$.

Explicit lists of minimal examples (e.g., certain graded upper-triangular matrix subalgebras, Grassmann algebras, group algebras with involution) define the boundary between polynomial and exponential growth; including one such algebra in a variety forces growth to be exponential (Cota et al., 4 Dec 2025, Cota et al., 5 Dec 2025).

3. Classification and Minimality for Quadratic and Low-Degree Polynomial Growth

Extensive classification results exist for varieties with exactly quadratic growth, and to a lesser extent, higher polynomial degrees. Minimal varieties of quadratic growth—those where every proper subvariety has strictly lower degree—are fully classified for group-graded algebras, $(G,*)$-algebras (graded algebras with homogeneous involution), and other structures (Cota, 5 Dec 2025, Cota et al., 5 Dec 2025).

The minimal generators for quadratic growth are finite-dimensional algebras with radical nilpotency index $3$ (e.g., commutative subalgebras of $UT_3$, Grassmann subalgebras with two generators and specific gradings/involutions, small block upper-triangular algebras). In group-graded settings, building blocks include $C_3^g$ (commutative subalgebras), $K_7^{g, h}$ (hook algebras), $\mathcal{G}_2^{g, h}$ (Grassmann), or certain $4 \times 4$ rank-one examples, and every quadratic-growth variety is a direct sum of these plus one-dimensional factors.

Minimality is sharply characterized: for each such block, any proper subvariety is nilpotent or of lower polynomial degree, and no two distinct minimal varieties coincide as $T$-ideals.

Examples using combinatorics of infinite words reveal the existence of uncountably many almost nilpotent nonassociative metabelian varieties with at most quadratic growth, exploiting Sturmian word complexity to control codimensions: for slopes $\alpha$ irrational, this yields $O(n^2)$ but not $O(n)$ growth (Mishchenko et al., 2017).

4. Polynomial Growth in Generalized Identities and Additional Structures

For $W$-algebras (algebras with an action of another algebra $W$ via multipliers), polynomial codimension growth is governed by the absence of specific exponential-growth minimal algebras (e.g., $UT_2$ with certain $W$-actions). The generalized exponent aligns with the ordinary G-exponent, and structural techniques—such as the Wedderburn–Malcev decomposition compatible with the $W$-action—remain central (Martino et al., 18 Feb 2025, Busalacchi et al., 27 Nov 2025).

In the superalgebra setting, classification results encompass varieties of $Z_2$-graded algebras. The only supervarieties with almost polynomial growth (exponential but all proper subvarieties polynomial) are those generated by the Grassmann algebra, $UT_2$ (trivial or elementary grading), or $D = F \oplus F$ with specific gradings. Analysis of central codimensions and cocharacters further refines the classification of subvarieties and their growth (Vieira et al., 13 Nov 2025).

Trace identities and varieties with trace also display dichotomic behavior. For a finite-dimensional algebra with trace, growth is polynomial if and only if the semisimple part is $F^k$ and the trace vanishes on the radical. The degree of the polynomial is the nilpotency index of the radical, with varieties generated by, e.g., $UT_2$ with zero trace, forming the minimal nontrivial polynomial-growth cases (Ioppolo et al., 2020).

5. Polynomial Growth in Finitely Generated Groups

In geometric group theory, groups of polynomial growth—those for which the size $|B_S(n)|$ of a Cayley graph ball grows as $O(n^d)$—are precisely the virtually nilpotent groups, with the degree $d$ determined by the Bass–Guivarc’h formula (Lyons et al., 2020). Recent results establish optimal explicit lower bounds: there exists $\varepsilon_d > 0$ depending only on the degree so that $|B_S(n)| \geq \varepsilon_d n^d$ for all $n$ and any Cayley graph of degree $d$. These bounds are completely effective, with minimal constants computed in terms of factorial data and double-exponential expressions. Such results yield isoperimetric consequences and universal gaps in percolation thresholds for graphs of polynomial growth.

6. Pathologies and Nonreduced Schemes

A critical distinction arises between reduced and nonreduced schemes or rings. On reduced projective schemes, the dimension growth functions for graded linear series are always asymptotically polynomial, and often exact up to arithmetic sequences or even globally. However, on nonreduced schemes, “pathological” behaviors are possible: the dimension function can oscillate between different polynomials with no limiting ratio $h^0(X, L_n)/n^d$, and the failure can persist on every arithmetic subsequence. Similar nonpolynomial oscillations arise in local nonreduced rings for lengths of quotients by graded families of $m$-primary ideals (Cutkosky, 2012).

This suggests that polynomial growth theory is only well behaved in the context of sufficiently reduced, “geometric” objects; nilpotents can destroy asymptotic regularity entirely.

7. Broader Context, Dichotomies, and Applications

The dichotomy between polynomial and exponential growth is a unifying theme. In PI-theory, in group theory, and for algebras with additional structures (traces, involutions, gradings, module algebras), sharp minimality criteria separate polynomially bounded from exponentially growing varieties. Almost polynomial growth—where a variety is just at the boundary—is realized by finitely many minimal examples, with every proper subvariety exhibiting only polynomial growth (Cota et al., 4 Dec 2025, Martino et al., 18 Feb 2025).

Classification techniques depend on reductions to semisimple and radical components, representation theory of symmetric or hyperoctahedral groups for cocharacter analysis, and combinatorial or convex-geometric constructions (Newton–Okounkov bodies). Explicit formulas and complete lists of minimal growth varieties have been established for many structural settings, though the full landscape for higher-degree polynomial growth remains a subject of continued investigation.

A plausible implication is that for each algebraic structure possessing a theory of polynomial identities and suitably finite simplicity, a finite explicit list of minimal exponential-growth obstructions controls the transition between polynomial and nonpolynomial growth, with much of the fine structure encoded by the radical and its action on semisimple components.