Lunardon-Polverino-Type Binomials

• Lunardon-Polverino-type binomials are Fq-linearized polynomials that serve as the foundation for constructing maximum scattered linear sets in projective geometry.
• They are characterized by an explicit algebraic structure and a norm condition that ensures scatteredness and projective uniqueness in PG(1, q^n).
• Classification results show that for large primes and field sizes, LP-type binomials uniquely satisfy the stringent criteria for scatteredness compared to other binomial forms.

A Lunardon-Polverino-type binomial is a specific $\mathbb{F}_q$-linearized polynomial, fundamental in the theory of scattered linear sets, particularly in finite geometry. Scattered binomials of this type, introduced by Lunardon and Polverino, play a central role in classifying and constructing maximum scattered linear sets in $\mathrm{PG}(1,q^n)$. The defining feature of LP-type binomials is their explicit algebraic structure and their relation to norm conditions over field extensions, underpinning their scatteredness properties and projective uniqueness within certain dimensional and field size constraints.

1. Definition and Scatteredness Condition

Let $q$ be a prime power and $n \geq 1$ an integer. An $\mathbb{F}_q$-linearized polynomial on $\mathbb{F}_{q^n}$ is of the form

$$f(X) = \sum_{i} a_i X^{q^i}, \quad a_i \in \mathbb{F}_{q^n}.$$

Such a polynomial is called scattered if, for all $x, y \in \mathbb{F}_{q^n}^*$,

$$\frac{f(x)}{x} = \frac{f(y)}{y} \implies \frac{x}{y} \in \mathbb{F}_q.$$

Equivalently, this scatteredness corresponds to the property that every affine $\mathbb{F}_{q^n}$-point of the curve

$$C_f: F(x, y) := x\left(y^{q^I} + a y^{q^J}\right) - y\left(x^{q^I} + a x^{q^J}\right) - (x^{q^I}y - x y^{q^I}) = 0$$

lies on an $\mathbb{F}_q$-rational line $x - p y = 0$ (Bartoli et al., 17 Feb 2025).

2. The Structure and Parameters of LP-Type Binomials

The LP family of scattered binomials is defined for integers $n \geq 4$, $1 \leq s \leq n-1$ with $\gcd(s, n) = 1$, and $\delta \in \mathbb{F}_{q^n}^*$. The binomial is given by

$f_s(x) = \delta x^{q^s} + x^{q^{n-s}}.$

Associated with $f_s$ is the $\mathbb{F}_q$-linear set in projective space,

$$U_{2,s} = \{ (x, f_s(x)) : x \in \mathbb{F}_{q^n} \} \subset \mathbb{F}_{q^n}^2,$$

and the corresponding projective linear set in $\mathrm{PG}(1, q^n)$,

$$L(U_{2, s}) = \{ \langle (x, f_s(x)) \rangle : x \in \mathbb{F}_{q^n}^* \}.$$

3. Necessary and Sufficient Conditions for Scatteredness

The primary characterization for the scatteredness of LP-type binomials is as follows:

• For $f(x) = x^{q^I} + a x^{q^J}$ on $\mathbb{F}_{q^n}$, $f$ is scattered if and only if

$$I + J = n, \quad \text{and} \quad N_{F_{q^n}/F_q}(a) \neq 1,$$

where $N_{F_{q^n}/F_q}$ denotes the norm from $\mathbb{F}_{q^n}$ to $\mathbb{F}_q$.

• In LP notation, $s + (n-s) = n$ and $N_{q^n/q}(\delta) \neq 1$.
• Theorem 3.9 (Bartoli et al., 17 Feb 2025) asserts that for $n \geq 5$, $\gcd(J-I, n) = 1$, $q > 2^{3n+4}$, $a \neq 0$,

$f(x) = x^{q^I} + a x^{q^J}$

is scattered if and only if $I+J = n$ and $N_{F_{q^n}/F_q}(a) \neq 1$.

4. Projective Uniqueness and Classification Results

For large $q$ and $n$ prime, the only scattered binomials are of LP-type:

• Corollary 3.10 (Bartoli et al., 17 Feb 2025): If $n \geq 5$ is prime and $q > 2^{3n+4}$, every scattered binomial on $\mathbb{F}_{q^n}$ is of LP-type.

For $n \leq 8$ and large $q$, a full classification is established (Theorem 4.10 (Bartoli et al., 17 Feb 2025)):

• LP family appears for all $n$ where $\gcd(I, n) = 1$, $I + J = n$, and $N_{q^n/q}(a) \neq 1$.
• For $n = 6, 8$, additional Csajbók–Marino–Polverino–Zanella (CMPZ)-type exceptions arise, but when $n$ is prime or not divisible by $2, 3, 4$, only LP-type exists.

For $\mathrm{PG}(1, q^4)$, LP-type binomials and the pseudoregulus examples exhaust the possibilities for maximum scattered $\mathbb{F}_q$-linear sets, and projective orbits are fully classified (Csajbók et al., 2017). The key invariants are $\alpha = b^{q^2+1} \in \mathbb{F}_{q^2}^*$ and $\beta = N_{\mathbb{F}_{q^4}/\mathbb{F}_q}(b)$, with each orbit determined by $(\alpha, \beta)$.

5. Connections to Linear Sets and Geometric Structure

The LP-type scattered binomials generate maximum scattered $\mathbb{F}_q$-linear sets in $\mathrm{PG}(1, q^n)$. Specifically, for $t \geq 4$,

$$U(b,t) = \left\{ (u, b u^{q} + u^{q^{t-1}}) : u \in \mathbb{F}_{q^t} \right\}$$

defines a maximum scattered linear set if and only if $N_{\mathbb{F}_{q^t}/\mathbb{F}_q}(b) \neq 1$ (Csajbók et al., 2017). The difference with pseudoregulus type is that the LP-type binomials are not $\mathbb{F}_{q^2}$-linear; the pseudoregulus example arises for $b = 0$, i.e., $f(x) = x$.

The uniqueness theorem shows that, up to $\mathrm{PGL}(2, q^4)$-equivalence, every non-pseudoregulus maximum scattered linear set in $\mathrm{PG}(1, q^4)$ is obtained from an LP-type binomial (Csajbók et al., 2017). The $\mathrm{GL}$-class of these sets is 1, meaning there is a unique $\mathbb{F}_q$-subspace (up to group action) for each linear set.

6. Proof Techniques and Algebraic Methodology

Classification results for LP-type binomials rely on:

• Analyzing the absence of nontrivial solution points (with $\frac{f(x)}{x} = \frac{f(y)}{y}$ and $x/y \notin \mathbb{F}_q$) on algebraic curves defined by $f$.
• Applying quadratic transformations and mapping affine conditions to the study of absolutely irreducible varieties in projective spaces.
• Using the Lang–Weil bound (Theorem 2.6 (Bartoli et al., 17 Feb 2025)) to estimate the number of $\mathbb{F}_q$-rational points and exclude non-LP binomials for sufficiently large $q$.
• Employing projective and semilinear equivalence criteria via explicit norm invariants (Csajbók et al., 2017).

Key results such as the adjoint criterion (Theorem 2.3 (Bartoli et al., 17 Feb 2025)) ensure that the scattered property is preserved under adjunction. Uniqueness of the LP family is established by reduction to canonical forms via collineation and geometric arguments involving Klein quadrics.

7. Summary Table: LP-Type Binomial Scatteredness

Condition	Scatteredness	Source
$N_{F_{q^n}/F_q}(a) \neq 1$	$f(x) = x^{q^{n-s}} + a x^{q^s}$ is scattered	(Bartoli et al., 17 Feb 2025)
$b = 0$	Pseudoregulus, scattered	(Csajbók et al., 2017)
$n \geq 5$ prime, $q > 2^{3n+4}$	Only LP-type possible	(Bartoli et al., 17 Feb 2025)
$t=4$, $\mathbb{F}_q$-linear set	Only pseudoregulus or LP-type	(Csajbók et al., 2017)

The scattered property and classification of LP-type binomials are critical for understanding the structure of linear sets in projective geometry, and the LP parameters and norm condition provide a complete algebraic criterion for their characterization for large fields and small dimensional cases.