Let $k=\mathbb{F}_q$, $E=\mathbb{F}_{q^n}$ and $\mathrm{Tr}=\mathrm{Tr}_{E/k}$. For $r\ge 2$, $a\in k^{\times}$ and $x\in E^{\times}$, let $\mathrm{N}(E,r,x,a)$ be the number of $r$-tuples $(x_1,\cdots,x_r)$ in $(E^{\times})^r$ satisfying $x_1\cdots x_r=x$ and $\mathrm{Tr}(x_1+\cdots+x_r)=a$. We prove $\left|\mathrm{N}(E,r,x,a)-\left((q^n-1)^{r-1}+(-1)^r\right)/q\right|\le (r^n-1) q^{\frac{(r-1)n-1}{2}}$. This proves the square-root estimate predicted in Wan's conjecture and generalizes a previous result of Moisio and Wan. For a finite semisimple algebra $B=\prod\limits_{i=1}^s M_{d_i}(\mathbb{F}_{q^{n_i}})$ over $k$ and a regular element $x\in B^{\times}$, the same method combined with Zelingher's formula leads to analogous square-root estimates.
Let $\mathbb{D}$ be a division ring with an involution $x \mapsto x^\star$, and $n \geq 2$ be an integer. Denote by $\mathcal{H}_n(\mathbb{D})$ the set of all $n$-by-$n$ Hermitian matrices with entries in $\mathbb{D}$, and by $\mathcal{A}\mathcal{H}_n(\mathbb{D})$ the set of all matrices $A-A^\star$ with $A \in \mathca...
Let $\Omega$ be a lattice in $\mathbb{C}$ with algebraic invariants and complex multiplication, let $\mathcal{E}$ be the elliptic curve associated with $\Omega$, and let $\wp$ be the Weierstrass function relative to $\Omega$. Set $ k:=\operatorname{End}(\mathcal{E}) \otimes_{\mathbb{Z}}\mathbb{Q}.$ We prove that if $t_...
Let $p$ be a prime and $\mathbb{F}_q/\mathbb{F}_r$ a finite field extension with $q=p^m$ and $r=p^s$. For any $k\mid q-1$, we consider $r$-ary irreducible cyclic codes (ICC) of the form $C(k,q/r) = \{(Tr_{q/r}(\gamma \omega^{ik})_{i=0}^{n-1})\}_{\gamma \in \mathbb{F}_r}$, with $\omega$ a primitive element of $\mathbb{F...
Let $\delta\in\mathbb{F}_{2^n}$ satisfy $\operatorname{Tr}_{\mathbb{F}_{2^n}/\mathbb{F}_2}(\delta)=1$. We study the permutation behavior of $$ f(x) = \left(\frac{1}{x^2+x+\delta}\right)^{2^k}+x $$ over $\mathbb{F}_{2^n}$. Helleseth and Zinoviev proved that $f(x)$ is a permutation for $k=0,1$, and remarked that numerica...
Let $W=W(B_n)$ act diagonally on $\mathfrak{h}\oplus\mathfrak{h}^*$, let $S=\mathbb{C}[\mathfrak{h}\oplus\mathfrak{h}^*]$, let $J\subset S$ be the ideal generated by the $W$-alternating polynomials and $\mathfrak{m}_S$ is the maximal ideal of the origin. For sufficiently large $m$ we compute $q,t$-Fuss-Catalan polynomi...
Let $V$ be an $n$-dimensional vector space over a finite field of order $q$. Let $r\geq 3$, $(r-1)n\geq rk$ and let $\mathcal F_1,\ldots,\mathcal F_r\subset \genfrac{[}{]}{0pt}{}{V}{k}$, where $\genfrac{[}{]}{0pt}{}{V}{k}$ denotes the set of $k$-dimensional subspaces of $V$. Suppose that $F_1\cap\cdots\cap F_r\neq\{0\}...
Toshihiro Shimizu, N. Tokushige· 1 citation
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