Let $k=\overline{\mathbb{F}_2}$. We construct a nine-dimensional $F$-rational affine variety $X$ admitting a locally closed embedding $X\hookrightarrow \mathbb{P}_k^{19}$ together with a dense open set $U\subseteq (\mathbb{P}^{19})^\vee$ such that $X\cap H$ is not $F$-rational (or even $F$-injective) for all $H\in U$. We also construct a nine-dimensional projective $F$-rational variety $\mathfrak{X}$ and a closed embedding $\mathfrak{X}\hookrightarrow\mathbb{P}^{29}_k$ under which the same conclusion holds.
Let $K$ be a number field and $S$ a finite set of non-archimedean places. Write $\mathcal{O}_S$ for the ring of $S$-integers of $K$ and $\mathcal{O}_S^\times$ for its unit group. Let $\pi : X \rightarrow \mathbb{P}^1$ be a morphism of (irreducible) curves defined over $K$, and denote by $\operatorname{Red}(\pi)$ the se...
For $a$ in $\mathbb{R}$, a subset $V$ contained in $\mathbb{R}^{n}$ is said to be $a$-convex if $x, y \in V \Longrightarrow a x + (1-a) y \in V$. According to Pinch [Math. Proc. Cambridge Philos. Soc., 1985], the $a$-convex hull of $V$ is the intersection of all of the $a$-convex subsets of $\mathbb{R}^{n}$ that contai...
Let $f$ be a regular endomorphism of $\mathbb{A}_{\mathbb{C}}^N$ and let $C\subseteq\mathbb{A}_{\mathbb{C}}^N$ be an irreducible curve. Suppose $C$ has an infinite intersection with the $f$-orbit of a point $x\in\mathbb{A}^N(\mathbb{C})$. Then the normalization of $C$ is isomorphic to either $\mathbb{A}^1$ or $\mathbb{...
Fix integers $k,d\geq2$. Let $\mathrm{End}_d^k$ denote the parameter space of endomorphisms of $\mathbb{P}^k$ of algebraic degree $d$. We prove that there exists a dense Zariski open subset $U_{d,k}\subset\mathrm{End}_d^k$, defined over $\mathbb{Q}$, such that, for every $f\in U_{d,k}(\mathbb{C})$, every dominant ratio...
Given a collection of algebraic numbers $\mathcal{S}\subset \overline{\mathbb{Q}}$ we study the varieties $V$ in $\mathbb{A}^m_\mathbb{C}$ such that $V(\mathbb{C})\cap\mathcal{S}^m$ is Zariski-dense in $V$. We show that for many classical families of algebraic numbers $\mathcal{S}$---such as the family of roots of gene...
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...
Yi Li· 0 citations
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