$r$-fat linearized polynomials over finite fields

Planar Polynomials arising from Linearized polynomials

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.

On the number of roots of some linearized polynomials

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 characterization of linearized polynomials with maximum kernel

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,…

Twisted Linearized Reed-Solomon Codes: A Skew Polynomial Framework

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,…

散射线性化多项式的标准形式及相关平移平面的性质

A standard form for scattered linearized polynomials and properties of the related translation planes

本文研究了散射线性化多项式稳定子群的结构，证明了其矩阵可同时对角化，得出了多项式的标准形式，并分析了相关平移平面存在非平凡仿射同调的充要条件及与广义 André 平面的关系。

Fattening up Warning's Second Theorem

We present a generalization of Warning's Second Theorem to polynomial systems over a finite local principal ring with suitably restricted input and output variables. This generalizes a recent result with Forrow and Schmitt (and gives a new proof of that result). Applications to additive group theory, graph theory and polynomial interpolation are pursued in detail.

Cancelation free formula for the antipode of linearized Hopf monoid

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 …

Linearized trinomials with maximum kernel

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…

On certain linearized polynomials with high degree and kernel of small dimension

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 w…

有限域上的对称张量分解

Symmetric Tensor Decompositions over Finite Fields

本文研究有限域扩张上乘法的对称张量秩，利用线性化多项式将其转化为span问题，并通过Frobenius自同构提出计算有效准则。

新的散射线性化四项式

New scattered linearized quadrinomials

本文研究有限域上线性化多项式的散射性质，构造并证明了一类新的四项式在特定参数下为部分散射或散射多项式，且相对于已知三类族是新的，其相关线性集合的 ΓL-类至少为二。

Combinatorics and geometry of power ideals

We investigate ideals in a polynomial ring which are generated by powers of linear forms. Such ideals are closely related to the theories of fat point ideals, Cox rings, and box splines. We pay special attention to a family of power ideals that arises naturally from a hyperplane arrangement A. We prove that their Hilbert series are determined by the combinatorics of A, and can be computed from its Tutte polynomial. We also obtain formulas for the Hilbert series of the resulting fat point ideals and zonotopal Cox rings. Our work unifies and generalizes results due to Dahmen-Micchelli, Holtz-Ron…