Quadratic Codimension Growth

• Quadratic codimension growth is defined by the quadratic increase in the dimension of multilinear components in algebras and projective schemes, leading to exact combinatorial bounds.
• It enables sharp classification and rigidity results in PI-algebras and operads, with exemplars such as twisted cubics, upper-triangular matrices, and the Grassmann algebra.
• Applications span projective geometry and graded/involutive algebras, employing precise asymptotic techniques to restrict degrees and classify minimal components.

Quadratic codimension growth refers to the phenomenon where the complexity of polynomial identities in a finite-dimensional associative algebra, or the dimension of the relevant multilinear components in an operad or algebraic variety, grows quadratically with the degree. This property arises naturally in several domains, notably in the geometry of schemes defined by quadratic equations, the representation theory of PI-algebras, and the paper of operadic and graded-identity growth. Despite the diversity of contexts, the analysis of quadratic codimension growth yields precise structural and classification results, as well as explicit combinatorial and asymptotic bounds.

1. Quadratic Codimension Growth in Projective Geometry

Let $X \subset \mathbb{P}^r$ be a complex projective scheme defined by a linear system $A\subset H^0(\mathbb{P}^r,\mathcal{O}(2))$ of quadrics, where $X$ is the base locus of $A$. Write $n = \dim X$, $c = \operatorname{codim} X = r-n$, and $d = \deg X$. The key algebraic–geometric problem is to bound $d$ as a function of $c$ when $I_X$ is generated by quadrics and certain geometric or syzygetic hypotheses are satisfied.

A fundamental theorem establishes that if $X$ is reduced in codimension $0$ and the locus $W$ of lines in $\mathbb{P}^r$ double-covered by the rational map defined by $A$ satisfies $\dim W \leq 2n+1$, then: $\binom{d}{2} \leq \binom{2c-1}{c-1}\;.$ Consequently, $d = O(2^c / c^{1/4})$ by Stirling's approximation, a strict improvement over the naive bound $d \leq 2^c$, which comes from monomial counting. If $X$ also enjoys property $N_p$ or $N_{2,p}$ (strengthened syzygy properties), a sharper result holds: $\binom{d+2-p}{2} \leq \binom{2c+3-2p}{c+1-p}\;,$ with corresponding improved asymptotics $d = O(2^c / c^{(p-1)/2})$.

Rigidity results classify schemes attaining equality: in the minimal codimensions, only the twisted cubic in $\mathbb{P}^3$ and the elliptic normal quintic in $\mathbb{P}^4$ realize the bound. In the presence of $N_p$ syzygies, the only extremal examples are the rational normal curve and elliptic quintic families. Thus, quadratic codimension growth in this context explicitly constrains both possible degrees and the geometric structure of $X$ (Alzati et al., 2010).

2. Structure of PI-Algebras and Varieties of Quadratic Growth

For a finitely-generated associative algebra $A$ over a field of characteristic zero satisfying a polynomial identity, the codimension sequence $c_n(A)$—the dimension of $n$-variable multilinear polynomials modulo the identities of $A$—serves as a central numerical invariant. In the PI-theory context, $c_n(A)$ can have polynomial (e.g., $n^t$) or exponential growth, depending on the structure of $A$.

Kemer theory provides a structure theorem: in a basic PI-algebra,

$c_n(A) = \Theta(cn^t d^n),$

with $d$ the PI-exponent (sum of squares of block sizes in the semisimple part), $q$ the number of simple components, and $s+1$ the Jacobson radical’s nilpotency index. The "polynomial part" exponent is $t = \frac{d-q}{2} + s$ (Aljadeff et al., 2015).

Specializing to quadratic growth $(t=2)$ yields a stringent algebraic classification: the only basic algebras with $c_n(A) = \Theta(n^2)$ are those where $A/J(A) \cong F^q$, $J(A)^3 = 0 \neq J(A)^2$, i.e., the semisimple quotient is a sum of $q$ copies of $F$, and the radical is nonzero but cubed radical vanishes. For example, the algebra of $3 \times 3$ upper-triangular matrices and the local algebra $F\langle x, y \rangle / (x, y)^3$ have this property.

3. Quadratic Codimension Growth in Graded and Involutive Algebras

In the graded and involutive setting, consider a finite group $G$ and a unitary $G$-graded algebra $A$ with a graded involution. The graded-involution codimension sequence $c_n^\#(A)$ counts multilinear $(G,*)$-polynomials not vanishing identically on $A$. Quadratic codimension growth in this context appears when $c_n^\#(A) = an^2 + bn + O(1)$.

A complete structural classification has been established: up to $T_G^*$-equivalence, every such algebra with quadratic growth is a direct sum of minimal unitary $(G,*)$-algebras exhibiting genuine quadratic growth in their codimension sequences. These include nilpotent-commutative algebras of index $2$ or $3$, $3 \times 3$ upper-triangular matrix subalgebras, certain two-generator Grassmann subalgebras with three involutive types, and commutative $4$-dimensional algebras with distinct involutions and gradings. The exact growth for each component is $c_n^\#(B) = a_B n^2 + b_B n + 1$ with $a_B \in\{0, \tfrac{1}{2}, 1\}$, and the sum over all components determines the total quadratic coefficient (Cota et al., 25 Nov 2025). Removal of any minimal component reduces the growth below quadratic.

4. Operadic and PI-Theoretic Unique Realization

For varieties of associative algebras (equivalently, quotients of the unital associative operad), the polynomial codimension sequence admits a unique realization for quadratic growth. In the operadic language, the only symmetric operad quotient of $\operatorname{uAss}$ with Gelfand–Kirillov dimension $3$ (grade $2$) is the quotient by the third truncation ideal $T_3$ (generated by the left-normed commutator $[x_1,[x_2,x_3]]$). The codimension sequence is: $c_n = n^2 - n + 1$ for all $n\geq 0$. This operadic variety is generated precisely by the infinite Grassmann (exterior) algebra, and every proper subvariety grows strictly slower. Thus, there is a unique nontrivial operadic PI-variety with quadratic codimension growth, rooted in the structure of the Grassmann algebra (Bao et al., 27 Apr 2025).

5. Algorithmic and Asymptotic Aspects

The problem of classifying all quadratic-growth operads and varieties is sharp: for binary quadratic operads with a finite number of generators and quadratic relations, the set of possible codimension sequences is infinite, and algorithmic determination of the growth exponent is undecidable for sufficiently many generators. However, for polynomially bounded (in particular, quadratic) codimension growth in symmetric or non-symmetric operads, the generating series for codimensions is always rational and the sequence is linear-recurrent. In the quadratic case, the generating function is explicitly computable, often rational, and the asymptotic is $an^2 + \cdots$ with a computable $a>0$ if and only if the operad is (operadically) equivalent to the Grassmann case (Piontkovski, 2017).

6. Consequences and Rigidity

Quadratic codimension growth imposes severe restrictions on algebraic, geometric, and operadic structures. In projective geometry, it bounds degrees and classifies extremal schemes. In PI-theory, it isolates unique basic algebras and operadic varieties attaining quadratic growth. The minimal components generating quadratic growth are structurally rigid: the inclusion of any additional relations, or exclusion of a minimal component, forces strictly subquadratic (linear or constant) growth. This establishes an isolated "quadratic growth stratum" both in geometric and algebraic classification landscapes.

7. Summary Table: Quadratic Codimension Growth Classifications

Context	Classification Criterion	Unique Quadratic Example
Projective schemes (Alzati et al., 2010)	Degree bound $d = O(2^c / c^{1/4})$ under quadric syzygies	Twisted cubic, elliptic normal quintic
Basic PI-algebras (Aljadeff et al., 2015)	$t = 2 \implies d = q,\, J^3=0 \neq J^2$, $A/J \cong F^q$	Upper-triangular $3\times3$ matrices, local algebra $\langle x,y\rangle/(x,y)^3$
Graded-involutive algebras (Cota et al., 25 Nov 2025)	Direct sum of minimal $(G,*)$-algebras with quadratic $c_n^\#$	Grassmann, nilpotent commutative, $3\times3$ triangular
Operadic PI-ideals (Bao et al., 27 Apr 2025)	Grade 2, $T_3$ ideal in $\operatorname{uAss}$	Grassmann (exterior) algebra

The above table underscores the unicity and rigidity of the quadratic codimension growth regime across algebraic, geometric, graded, and operadic categories.