arXiv subjects

Negin Shadgar

Publications and source records attributed to Negin Shadgar.

3 recordsLinked to original sources

arXiv2026

Collision Parameters in Higher-Dimensional Families of Polynomials

Let $d\geq 3$ and consider the generic family of monic normalized polynomials $f_{\mathbf{t}}(z)=z^d+t_1z^{d-2}+t_2z^{d-3}+\cdots+t_{d-2}z+t_{d-1}$, where $\mathbf{t}=(t_1,\ldots,t_{d-1})$. Given pairwise distinct starting points $α_1,\ldots,α_d\in\bar{\mathbb{Q}}$ and a target point $β\in\bar{\mathbb{Q}}$, we show that the set of parameters $\boldsymbolλ\in\mathbb{A}^{d-1}(\bar{\mathbb{Q}})$ for which $β$ belongs to the forward orbit of every $α_i$ under $f_{\boldsymbolλ}$ is not Zariski dense in $\mathbb{A}^{d-1}$.

math.NT↗

arXiv2026

Collision of Orbits for Families of Polynomials Defined over Number Fields

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 $f_λ(z):=z^d+\sum_{i=0}^{d-2} c_i(λ)\cdot z^i$ parameterized by $λ\in\bar{\mathbb{Q}}$; 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 $α_1(t),α_2(t),β(t)\in\bar{\mathbb{Q}}[t]$, where $α_i(t)$ 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 $λ\in\bar{\mathbb{Q}}$ with the property that for some $m,n\in\mathbb{N}$ (depending on $λ$), we have that $f_λ^m(α_1(λ))=f_λ^n(α_2(λ))=β(λ)$.

math.NT↗

arXiv2026

Collision of Orbits on an Elliptic Surface

Let $C$ be a smooth projective curve defined over $\Qbar$, let $π:\mathcal{E}\lra C$ be an elliptic surface and let $σ_{P_1},σ_{P_2},σ_{Q}$ be sections of $π$ (corresponding to points $P_1,P_2, Q$ of the generic fiber $E$ of $\mathcal{E}$). We obtain a precise characterization, expressed solely in terms of the dynamical relations between the points $P_1,P_2,Q$ with respect to the endomorphism ring of $E$, so that there exist infinitely many $ł\in C(\Qbar)$ with the property that for some nonzero integers $m_{1,ł},m_{2,ł}$, we have that $[m_{i,ł}](σ_{P_{i}}(ł))=σ_{Q}(ł)$ (for $i=1,2$) on the smooth fiber $E_ł$ of $\mathcal{E}$.

math.NT↗