Almost-Cubic Polynomials: Theory & Applications

• Almost-cubic polynomials are integer polynomials of degree at most three that form the basis of Diophantine approximation and cubic field arithmetic.
• They exhibit precise zero-infinity laws with explicit convergence criteria, impacting the determination of Hausdorff dimensions in metric number theory.
• Their rich algebraic structure connects Ramanujan cubics, the cubic Brahmagupta identity, and elliptic curve arithmetic to streamline computations in cubic fields.

Almost-cubic polynomials are central objects in the metric theory of Diophantine approximation and the arithmetic of cubic fields. The term refers to integer polynomials of degree at most three, and more generally to objects and theorems that parallel the cubic (degree three) setting in fields such as approximation, field arithmetic, and algebraic dynamics. This article surveys the core concepts, classification criteria, Diophantine implications, algebraic identities, and open problems connected with almost-cubic polynomials, drawing on the metric theory of Pezzoni (Pezzoni, 2018), the Ramanujan-cubic paradigm (Dresden et al., 2017), and the cubic Brahmagupta identity (Hambleton, 2018).

1. General Framework and Definitions

Given $n \ge 1$, let $$\mathrm{Polys}_n = \{ P \in \mathbb{Z}[X] : \deg P \le n \}$$ denote the family of integer polynomials of degree at most $n$. The height of %%%%3%%%% is $H(P) = \max\{ |a_i| : 0 \leq i \leq n \}$, and the discriminant $\mathrm{Disc}(P)$ is a homogeneous form of degree $2n-2$ in the coefficients, always satisfying $|\mathrm{Disc}(P)| \le C_n H(P)^{2n - 2}$ for some $C_n > 0$ depending only on $n$.

A monotonic decreasing function $\psi : (0, \infty) \to (0, \infty)$ is called an approximation function. For such $\psi$, the limsup set

$$A_n(\psi) = \left\{ x \in \mathbb{R} : |P(x)| \leq \psi(H(P)) \text{ for infinitely many } P \in \mathrm{Polys}_n \right\}$$

captures the real numbers $x$ approximable "almost by cubic" polynomials of controlled height. Restricting to polynomials with bounded discriminant, define, for $0<\lambda\leq n-1$,

$$\begin{aligned} \mathrm{Polys}_n^\lambda &= \left\{ P \in \mathrm{Polys}_n : |\mathrm{Disc}(P)| \gg H(P)^{2(n-1 - \lambda)} \right\}, \ \mathrm{Polys}_{n,\lambda} &= \left\{ P \in \mathrm{Polys}_n : |\mathrm{Disc}(P)| \ll H(P)^{2(n-1 - \lambda)} \right\}. \end{aligned}$$

Corresponding limsup sets $A_n^\lambda(\psi)$ and $A_{n,\lambda}(\psi)$ are defined analogously. These encapsulate the Diophantine behavior of numbers approximated by “almost-cubic” polynomials with certain extremal or typical discriminants (Pezzoni, 2018).

2. Metric Diophantine Approximation and Jarník-type Theorems

For $n=3$ (the cubic case), the primary questions concern the Hausdorff dimension and measure of $A_3(\psi)$ and its discriminant-restricted subfamilies. Given a dimension function $g$ (increasing, continuous, $g(r) \to 0$ as $r \to 0$) satisfying standard monotonicity properties, Pezzoni (Pezzoni, 2018) establishes the following laws:

• Divergence Law: If $\sum_{q=1}^\infty g(\psi(q)/q) q^3 = \infty$, then $\mathcal{H}^g(A_3(\psi)) = \infty$. For $\psi(q) = q^{-w}$ and $g(r) = r^s$, this yields a divergence of $\sum q^{-s(w+1)+3}$ precisely for $s \leq 4/(w+1)$, and thus $\dim_H A_3(w) \geq 4/(w+1)$.
• Convergence Laws:
  • For the bounded-derivative family $A_3^\lambda(\psi)$ ($0 < \lambda < 1$): if $\sum_{q=1}^\infty g(\psi(q)/q) q^3 < \infty$, then $\mathcal{H}^g(A_3^\lambda(\psi)) = 0$.
  • For irreducible polynomials with small discriminant ($0 \leq \lambda < 9/20$): if $$\sum_{q=1}^\infty g(\psi(q)/q) q^{3-2\lambda/3} < \infty$$, then $\mathcal{H}^g(A_{3,\lambda}^*(\psi)) = 0$ (where $A_{3, \lambda}^*(\psi)$ is defined on irreducibles in $\mathrm{Polys}_{3, \lambda}$).

Taking $\lambda=0$ recovers Bernik's exact dimension formula

$\dim_H A_3(w) = \frac{4}{w+1}.$

A complete zero-infinity law is thereby established for almost-cubic approximation, analogous to Jarník’s classical theorems for linear forms, both in terms of precise Hausdorff dimension and $g$-measure, with convergence criteria determined by explicit series.

3. Algebraic and Structural Characterization of Cubic Polynomials

Every monic cubic $f(x) = x^3 + A x^2 + B x + C$ with complex coefficients and non-repeated roots is either a translate of $y=x^3$ or can be reduced, by an affine change of variable, to a Ramanujan simple cubic (Dresden et al., 2017):

• Translate Case: $f(x) = (x + h)^3 + k$ if and only if $B = A^2/3$. Here, $h = A/3$, $k = C - A^3/27$.
• Non-Translate Case: For $B \neq A^2/3$, $f$ can be converted to a Ramanujan cubic $$p_{B'}(x) = x^3 - \frac{3+B'}{2} x^2 - \frac{3-B'}{2} x + 1$$ via the transformation $x \mapsto (a-x)/c$, where $a, c$ are explicit rational functions of $A, B, C$ and the discriminant $\Delta$.

The roots of any cubic can therefore be expressed in terms of the roots of an associated Ramanujan cubic, permitting uniform closed-form trigonometric and radical parametric representations.

Ramanujan cubics enjoy a rich symmetry: the map $n(x) = 1/(1-x)$ permutes their roots, reflecting a group-theoretic undercurrent in almost-cubic polynomial geometry.

4. Cubic Brahmagupta Identity and Arithmetic in Cubic Fields

A binary cubic form $C(x,y) = a x^3 + b x^2 y + c x y^2 + d y^3$ of discriminant $\Delta$, with root $\theta$ of $C(\theta, 1) = 0$, defines a cubic field $K = \mathbb{Q}(\theta)$. The integral basis $B = \{1, a \theta, a \theta^2 + b \theta\}$ enables unique representation of $\alpha \in \mathcal{O}_K$ as $u + x (a \theta) + y (a \theta^2 + b \theta)$ with $u,x,y \in \mathbb{Z}$ (Hambleton, 2018). The associated $3 \times 3$ multiplication matrix $N(\alpha)$ satisfies

$N(\alpha) N(\beta) = N(\alpha \beta),$

with norms and traces given by $\det N(\alpha)$ and $\mathrm{tr}\, N(\alpha) = 3u - bx - 2cy$. This generalizes the classical Brahmagupta–Pell identity to the cubic setting, unifying cubic field arithmetic with matrix algebra and enabling efficient computation of trace, norm, inversion, and composition.

A crucial Diophantine equation emerges: $t^3 - 3 t Q(x,y) + F(x,y) = 27 n,$ where $t$ is the trace, $n$ is the norm, and $Q$, $F$ are the covariants. For fixed $t$ and $n$, this defines a (possibly singular) cubic curve, which is generically an elliptic curve, forging deep links between cubic field elements and elliptic-curve arithmetic.

5. Proof Sketch for Convergence Criteria in Almost-Cubic Approximation

In the convergence case for $A_3^\lambda(\psi)$, the argument proceeds as follows (Pezzoni, 2018):

1. Restrict $x$ to a compact interval $I = [-1/2, 1/2)$ using translation invariance.
2. Partition $\mathrm{Polys}_3^\lambda$ by dyadic height intervals $H(P) \in [2^t, 2^{t+1})$.
3. For each $P$, define a sublevel set $$\sigma_\epsilon(P) = \{ x \in I : |P(x)| \leq \epsilon, |P'(x)| \geq 2 \}$$, where $\epsilon = \psi(2^t)$. These sets are efficiently covered due to a uniform derivative bound, itself a consequence of the fixed lower bound on discriminant $|\mathrm{Disc}(P)| \gg H(P)^{2(2-\lambda)}$.
4. Enlarging each interval to have length $\ell = \epsilon/2^t$, the total number of such intervals is $\ll 2^{4t}$. The $g$-measure of the covering is then bounded by $\sum_t 2^{4t} g(\psi(2^t)/2^t)$, converging precisely when $\sum_q g(\psi(q)/q) q^3$ does.
5. The covering argument on $I$ extends to all of $\mathbb{R}$ by translation.

These steps ensure that whenever the relevant series converges, the Hausdorff $g$-measure of $A_3^\lambda(\psi)$ vanishes.

6. Open Problems, Applications, and Connections to Elliptic Curves

Several open directions arise from the arithmetic of almost-cubic polynomials:

• Elliptic Curve Correspondence: The Diophantine locus defined by fixed trace and norm elements in a cubic field corresponds to rational points on associated elliptic curves. It remains open to classify precisely which elliptic curves arise in this way, given a cubic form $C$ and invariants $(t,n)$ (Hambleton, 2018).
• Norm-One Loci and Generalized Pell Equations: For trace $t=0$, norm $n=1$, the resulting curve is the genus-1 norm-one locus, whose integral points form a multiplicative group generalizing Pell's equation to cubic fields.
• Efficient Field Arithmetic: The cubic Brahmagupta identity provides a matrix-algebraic approach to arithmetic in cubic orders, streamlining computations of multiplication, trace, norm, and units.
• Algorithmic and Metric Implications: The sharp zero-infinity law governing approximation by almost-cubic polynomials has potential implications in algorithmic number theory and metric geometry of number fields.

These aspects highlight the multifaceted theory of almost-cubic polynomials as a crossroads of Diophantine approximation, algebraic number theory, and arithmetic geometry.