---
title: Lunardon-Polverino-Type Binomials
url: https://www.emergentmind.com/topics/lunardon-polverino-type-binomials
type: topic
---

# Lunardon-Polverino-Type Binomials

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 $x - p y = 0$ [2502.11666].

## 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 [2502.11666] 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 [2502.11666]: 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 [2502.11666]):
- 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 [1705.00731]. 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 $N_{\mathbb{F}_{q^t}/\mathbb{F}_q}(b) \neq 1$ [1705.00731]. 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 [1705.00731]. 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 [2502.11666]) 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 [1705.00731].

Key results such as the adjoint criterion (Theorem 2.3 [2502.11666]) 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 | [2502.11666]     |
| $b = 0$                                    | Pseudoregulus, scattered      | [1705.00731]     |
| $n \geq 5$ prime, $q > 2^{3n+4}$           | Only LP-type possible         | [2502.11666]     |
| $t=4$, $\mathbb{F}_q$-linear set           | Only pseudoregulus or LP-type | [1705.00731]     |

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.

Source: https://www.emergentmind.com/topics/lunardon-polverino-type-binomials