Let $d\ge 2$ be an integer and let $c_0(t),\dots, c_{d-2}(t)\in\bar{\mathbb{Q}}[t]$. We consider the family of normalized polynomials $f_\lambda(z):=z^d+\sum_{i=0}^{d-2} c_i(\lambda)\cdot z^i$ parameterized by $\lambda\in\bar{\mathbb{Q}}$; the generic element of our family of polynomials is $f_t(z):=z^d+\sum_{i=0}^{d-2}c_i(t)\cdot z^i\in \bar{\mathbb{Q}}[t][z]$. Also, let $\alpha_1(t),\alpha_2(t),\beta(t)\in\bar{\mathbb{Q}}[t]$, where $\alpha_i(t)$ is not preperiodic under the action of $f_t(z)$ for each $i=1,2$. Under some natural hypotheses, we obtain precise necessary and sufficient conditions for which there exist infinitely many $\lambda\in\bar{\mathbb{Q}}$ with the property that for some $m,n\in\mathbb{N}$ (depending on $\lambda$), we have that $f_\lambda^m(\alpha_1(\lambda))=f_\lambda^n(\alpha_2(\lambda))=\beta(\lambda)$.
For an integer $d\ge 3$, put $\Delta_d=\min{2^{d-1},d(d-1)}$. Let $a/q$ be reduced, let $P(X)=\frac{a}{q}X^d+\alpha_{d-1}X^{d-1}+\cdots+\alpha_0$, and let $\mathcal{I}$ be an interval of $H\le q$ consecutive integers. We prove $\left|\sum_{n\in\mathcal{I}}e(P(n))\right|\ll_{d,\varepsilon}q^{1/d}H^\varepsilon+H^{1-1/\De...
We study the approximation of the number of solutions of Laurent polynomials over finite fields. For a Laurent polynomial \[f(x)=\sum_{j=1}^{s}a_jx^{u_j}\in \mathbb{F}_q[x_1^{\pm1},\ldots,x_n^{\pm1}], \] let $U$ be its augmented support matrix whose columns are $(1,u_j)$ with rank $\rho$ and $N(f) := \# \{x\in (\mathbb...
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...
In this paper, we consider the asymptotic density of $T_c(x):=\#\{p\leq x:P^+(p-1)\geq p^c\}$, where $P^+(n)$ denote the largest prime factor of $n$. We show that for $x\rightarrow\infty$, for any $0<c\leq 1/2$, one has \begin{align*} \mathop{\lim\inf}_{x\rightarrow\infty}\frac{T_c(x)}{\pi(x)}&\geq\max\left(1-\frac{16}...
Let $X=\{X_j\}_{j=1}^\infty$ be a sequence of independent random variables whose densities and moments of order $2d$ are uniformly bounded. For a random vector $f(X)=(f_1(X),f_2(X))$ whose components are polynomial functionals of degree at most $d$, we prove that \[ [[f]]_{\mu,\infty}^{\frac1{2d-1}}\mu(f\in A) \le C\bi...
Let $f=(f_1,\dots,f_r)$ be a complete factorization of a central hyperplane arrangement $D$ in $X=\mathbb{C}^n$. For a monoid ideal $K\subseteq \mathbb{N}^r$ we study the Bernstein-Sato ideal $B^K_f$ of $f$ along $K$, that is, the $\mathbb{C}[s]$-annihilator of $\mathcal{D}_X[s]f^s/\sum_{m\in K}\mathcal{D}_X[s]f^{s+m}$...
Wen-Zong Guo, Lei Wu, Fang-Han Xiang· 0 citations
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