Let $f$ be the $\mathbb{F}_q$-linear map over $\mathbb{F}_{q^{2n}}$ defined by $x\mapsto x+ax^{q^s}+bx^{q^{n+s}}$ with $\gcd(n,s)=1$. It is known that the kernel of $f$ has dimension at most $2$, as proved by Csajbók et al. in "A new family of MRD-codes" (2018). For $n$ big enough, e.g. $n\geq5$ when $s=1$, we classify the values of $b/a$ such that the kernel of $f$ has dimension at most $1$. To this aim, we translate the problem into the study of some algebraic curves of small degree with respect to the degree of $f$; this allows to use intersection theory and function field theory together with the Hasse-Weil bound. Our result implies a non-scatteredness result for certain high degree scattered binomials, and the asymptotic classification of a family of rank metric codes.
Linearized polynomials have attracted a lot of attention because of their applications in both geometric and algebraic areas. Let $q$ be a prime power, $n$ be a positive integer and $σ$ be a generator of $\mathrm{Gal}(\mathbb{F}_{q^n}\colon\mathbb{F}_q)$. In this paper we provide closed formulas for the coefficients of a $σ$-trinomial $f$ over $\mathbb{F}_{q^n}$ which ensure that the dimension of the kernel of $f$ equals its $σ$-degree, that is linearized polynomials with maximum kernel. As a consequence, we present explicit examples of linearized trinomials with maximum kernel and characteriz…
In this paper we construct planar polynomials of the type $f_{A,B}(x)=x(x^{q^2}+Ax^{q}+Bx)\in \mathbb{F}_{q^3}[x]$, with $A,B \in \mathbb{F}_{q}$. In particular we completely classify the pairs $(A,B)\in \mathbb{F}_{q}^2$ such that $f_{A,B}(x)$ is planar using connections with algebraic curves over finite fields.
A linearized polynomial $f(x)\in\mathbb F_{q^n}[x]$ is called scattered if for any $y,z\in\mathbb F_{q^n}$, the condition $zf(y)-yf(z)=0$ implies that $y$ and $z$ are $\mathbb F_{q}$-linearly dependent. In this paper two generalizations of the notion of a scattered linearized polynomial are defined and investigated. Let $t$ be a nontrivial positive divisor of $n$. By weakening the property defining a scattered linearized polynomial, L-$q^t$-partially scattered and R-$q^t$-partially scattered linearized polynomials are introduced in such a way that the scattered linearized polynomials are preci…
We provide an algebraic description for sum-rank metric codes, as quotient space of a skew polynomial ring. This approach generalizes at the same time the skew group algebra setting for rank-metric codes and the polynomial setting for codes in the Hamming metric. This allows to construct twisted linearized Reed-Solomon codes, a new family of maximum sum-rank distance codes extending at the same time Sheekey's twisted Gabidulin codes in the rank metric and twisted Reed-Solomon codes in the Hamming metric. Furthermore, we provide an analogue in the sum-rank metric of Trombetti-Zhou construction,…
In this paper, the compositional inverses of a class of linearized permutation polynomials of the form $P(x)=x+x^2+\tr(\frac{x}{a})$ over the finite field $\mathbb{F}_{2^n}$ for an odd positive integer $n$ are explicitly determined.
We provide sufficient and necessary conditions for the coefficients of a $q$-polynomial $f$ over $\mathbb{F}_{q^n}$ which ensure that the number of distinct roots of $f$ in $\mathbb{F}_{q^n}$ equals the degree of $f$. We say that these polynomials have maximum kernel. As an application we study in detail $q$-polynomials of degree $q^{n-2}$ over $\mathbb{F}_{q^n}$ which have maximum kernel and for $n\leq 6$ we list all $q$-polynomials with maximum kernel. We also obtain information on the splitting field of an arbitrary $q$-polynomial. Analogous results are proved for $q^s$-polynomials as well,…
Cycle structures and parities of linearized permutation polynomials
KLMM, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；Hubei Province Key Laboratory of Applied Mathematics, Faculty of Mathematics and Statistics, Hubei University（湖北大学数学与统计学院）
Linearized polynomials appear in many different contexts, such as rank metric codes, cryptography and linear sets, and the main issue regards the characterization of the number of roots from their coefficients. Results of this type have been already proved in [7,10,24]. In this paper we provide bounds and characterizations on the number of roots of linearized polynomials of this form \[ ax+b_0x^σ+b_1x^{σq^n}+b_2x^{σq^{2n}}+\ldots+b_{t-1}x^{σq^{n(t-1)}} \in \mathbb{F}_{q^{nt}}[x], \] with $σ$ a generator of the Galois group $\mathrm{Gal}(\mathbb{F}_{q^n}\colon\mathbb{F}_q)$. Also, we characteri…
A linearized polynomial over $\mathbb F_{q^n}$ is called scattered when for any $t,x\in\mathbb F_{q^n}$, the condition $xf(t)-tf(x)=0$ holds if and only if $x$ and $t$ are $\mathbb F_q$-linearly dependent. General conditions for linearized polynomials over $\mathbb F_{q^n}$ to be scattered can be deduced from the recent results in [4,7,15,19]. Some of them are based on the Dickson matrix associated with a linearized polynomial. Here a new condition involving Dickson matrices is stated. This condition is then applied to the Lunardon-Polverino binomial $x^{q^s}+δx^{q^{n-s}}$, allowing to prove t…
For an arbitrary $q$-polynomial $f$ over $\mathbb{F}_{q^n}$ we study the problem of finding those $q$-polynomials $g$ over $\mathbb{F}_{q^n}$ for which the image sets of $f(x)/x$ and $g(x)/x$ coincide. For $n\leq 5$ we provide sufficient and necessary conditions and then apply our result to study maximum scattered linear sets of $\mathrm{PG}(1,q^5)$.
In this paper we prove that the property of being scattered for a $\mathbb{F}_q$-linearized polynomial of small $q$-degree over a finite field $\mathbb{F}_{q^n}$ is unstable, in the sense that, whenever the corresponding linear set has at least one point of weight larger than one, the polynomial is far from being scattered. To this aim, we define and investigate $r$-fat polynomials, a natural generalization of scattered polynomials. An $r$-fat $\mathbb{F}_q$-linearized polynomial defines a linear set of rank $n$ in the projective line of order $q^n$ with $r$ points of weight larger than one. W…
A class of scattered linearized polynomials covering infinitely many field extensions is exhibited. More precisely, the $q$-polynomial over $\mathbb F_{q^6}$, $q \equiv 1\pmod 4$ described in arXiv:1906.05611, arXiv:1910.02278 is generalized for any even $n\ge6$ to an $\mathbb F_q$-linear automorphism $ψ(x)$ of $\mathbb F_q^n$ of order $n$. Such $ψ(x)$ and some functional powers of it are proved to be scattered. In particular this provides new maximum scattered linear sets of the projective line $\mathrm{PG}(1,q^n)$ for $n=8,10$. The polynomials described in this paper lead to a new infinite f…
We study multivariable Christoffel-Darboux kernels, which may be viewed as reproducing kernels for antisymmetric orthogonal polynomials, and also as correlation functions for products of characteristic polynomials of random Hermitian matrices. Using their interpretation as reproducing kernels, we obtain simple proofs of Pfaffian and determinant formulas, as well as Schur polynomial expansions, for such kernels. In subsequent work, these results are applied in combinatorics (enumeration of marked shifted tableaux) and number theory (representation of integers as sums of squares).
Many combinatorial Hopf algebras $H$ in the literature are the functorial image of a linearized Hopf monoid $\bf H$. That is, $H={\mathcal K} ({\bf H})$ or $H=\overline{\mathcal K} ({\bf H})$. Unlike the functor $\overline{\mathcal K}$, the functor ${\mathcal K}$ applied to ${\bf H}$ may not preserve the antipode of ${\bf H}$. In this case, one needs to consider the larger Hopf monoid ${\bf L}\times{\bf H}$ to get $H={\mathcal K} ({\bf H})=\overline{\mathcal K}({\bf L}\times{\bf H})$ and study the antipode in ${\bf L}\times{\bf H}$. One of the main results in this paper provides a cancelation …
Many problems and conjectures in extremal combinatorics concern polynomial inequalities between homomorphism densities of graphs where we allow edges to have real weights. Using the theory of graph limits, we can equivalently evaluate polynomial expressions in homomorphism densities on kernels $W$, i.e., symmetric, bounded, and measurable functions $W$ from $[0,1]^2 \to \mathbb{R}$. In 2011, Hatami and Norin proved a fundamental result that it is undecidable to determine the validity of polynomial inequalities in homomorphism densities for graphons (i.e., the case where the range of $W$ is $[0…
Let $H$ be a fixed graph. Given a graph $G$ and an integer $k$, the $H$-free edge modification problem asks whether it is possible to modify at most $k$ edges in $G$ to make it $H$-free. Sandeep and Sivadasan (IPEC 2015) asks whether the paw-free completion problem and the paw-free edge deletion problem admit polynomial kernels. We answer both questions affirmatively by presenting, respectively, $O(k)$-vertex and $O(k^4)$-vertex kernels for them. This is part of an ongoing program that aims at understanding compressibility of $H$-free edge modification problems.
In the last years, kernelization with structural parameters has been an active area of research within the field of parameterized complexity. As a relevant example, Gajarsk{y} et al. [ESA 2013] proved that every graph problem satisfying a property called finite integer index admits a linear kernel on graphs of bounded expansion and an almost linear kernel on nowhere dense graphs, parameterized by the size of a $c$-treedepth modulator, which is a vertex set whose removal results in a graph of treedepth at most $c$, where $c \geq 1$ is a fixed integer. The authors left as further research to inv…
We study the existence of polynomial kernels, for parameterized problems without a polynomial kernel on general graphs, when restricted to graphs of bounded twin-width. Our main result is that a polynomial kernel for $k$-Dominating Set on graphs of twin-width at most 4 would contradict a standard complexity-theoretic assumption. The reduction is quite involved, especially to get the twin-width upper bound down to 4, and can be tweaked to work for Connected $k$-Dominating Set and Total $k$-Dominating Set (albeit with a worse upper bound on the twin-width). The $k$-Independent Set problem admits…
