arXiv subjects

Tomasz Ślusarski

Publications and source records attributed to Tomasz Ślusarski.

4 recordsLinked to original sources

arXiv2026

Polynomial representatives of finite-field maps: a sharp dimensional dichotomy

Let $k=\mathbb{F}_q$. A polynomial representative of a finite-set map is a tuple of polynomials inducing that map on the rational-point grid. We prove a sharp distinction between a finite-set map and the geometry of its representatives. If $n=1$ or $n=2$, every polynomial representative of a permutation of $k^n$ has algebraically independent coordinates. If $n\geq3$, every set map $k^n\to k^n$ has both an algebraically independent and an algebraically dependent representative; the latter may be chosen to satisfy \[ F_2^q-F_2=(F_1^q-F_1)F_3. \] More generally, every map $k^m\to k^n$ has an algebraically independent representative exactly when $n\leq m$, while every such map has a dependent representative when $n\geq3$. The dependent construction combines an Artin--Schreier interpolation theorem, producing prescribed values by polynomials $A,B$ with $A^q-A\mid B^q-B$, with a three-coordinate suspension. For the identity on $k^3$, the scheme-theoretic image may be chosen to be exactly \[ V^q-V=(U^q-U)W, \] a smooth geometrically integral rational surface. We also establish low-degree and extension-field criteria forcing algebraic independence. An exact exhaustive computation additionally proves that every $2$-reduced representative of a permutation of $\mathbb{F}_2^3$ has algebraically independent coordinates.

math.NT↗

arXiv2026

Iterated Cartier Flux and Non-Finite Generation of Tame Polynomial Automorphism Groups over Finite Fields

Let $k=\mathbb F_q$ be a finite field of characteristic $p$, and let \[ U_n(k)=\{F\in\operatorname{TA}_n(k):F(0)=0,\ JF(0)=I_n\}. \] We prove that every special tame polynomial automorphism is repeatedly Cartier-admissible for the de Rham flux class associated with \[ λ=x_1\,dx_2\wedge\cdots\wedge dx_n. \] Lowest-weight projection of the iterated Cartier descents gives additive characters \[ χ_{r,j}:U_n(k)\longrightarrow(k,+), \qquad r\ge0,\quad 1\le j\le n, \] and for every fixed group element all but finitely many of these characters vanish. Put \[ S_{n,p}=\begin{cases} \mathbf N_{\ge1},&(n,p)=(2,2),\\ \mathbf N_0,&\text{otherwise}. \end{cases} \] The joint character map is surjective onto \[ \bigoplus_{s\in S_{n,p}}(k^n,+), \] and every fixed-coordinate submap onto \[ \bigoplus_{s\in S_{n,p}}(k,+) \] admits an explicit group-theoretic section. Consequently, \[ \dim_{\mathbb F_p} \frac{U_n(k)}{[U_n(k),U_n(k)]\,U_n(k)^{[p]}} =\aleph_0. \] The character $χ_{r,j}$ factors through the jet of order \[ N_r=(n-1)(p^{r+1}-1), \] and this order is optimal for every $r\in S_{n,p}$. Since $U_n(k)$ has finite index in $\operatorname{TA}_n(k)$, the tame polynomial automorphism group over a finite field is finitely generated exactly in dimension one. In particular, this proves the Maubach--Willems finite-generation conjecture for $n\ge3$.

math.AG↗

arXiv2026

Prescribed extension spectra of mock automorphisms over finite fields

We classify the finite-extension permutation spectra occurring in an explicit family of mock automorphisms over finite fields. Every finite nonempty divisor-closed subset of $\N_{>0}$ occurs; in particular, this disproves Maubach and Willems conjecture that every mock automorphism permutes infinitely many finite extensions. Every finite-spectrum map satisfying \[ \Jac(F)=I_n,\qquad F|_{\F_q^n}=\id \] has total degree at least $p(q+1)$. For every $n\geq2$, this bound is attained by a map with spectrum $\{1\}$. In dimension one, the same bound and exact spectrum $\{1\}$ are attained over every odd field and over $\F_2$; for even $q>2$ we give a nonexceptional construction with an effective finite bound for its spectrum. If $δ_p$ denotes the least degree of a nonlinear reduced permutation polynomial over $\mathbb F_p$, then the least degree $μ_p$ of a nonexceptional one-variable mock automorphism satisfies \[ μ_2=6,\qquad μ_3=12,\qquad μ_p=pδ_p\quad(p\ge5). \] We also record a cofinite-spectrum criterion and isolate the surviving prime-to-$p$ question. Every nontrivial counterexample constructed here has geometric generic degree divisible by $p$.

math.AG↗

arXiv2023

Nonmonotonic buildup of spin-singlet correlations in a double quantum dot

Dynamical buildup of spin-singlet correlations between the two quantum dots is investigated by means of the time-dependent numerical renormalization group method. By calculating the timeevolution of the spin-spin expectation value upon a quench in the hopping between the quantum dots, we examine the time scales associated with the development of an entangled spin-singlet state in the system. Interestingly, we find that in short time scales the effective exchange interaction between the quantum dots is of ferromagnetic type, favoring spin-triplet correlations, as opposite to the long time limit, when strong antiferromagnetic correlations develop and eventually an entangled spin-singlet state is formed between the dots. We also numerically determine the relevant time scales and show that the physics is generally governed by the interplay between the Kondo correlations on each dot and exchange interaction between the spins of both quantum dots.

cond-mat.mes-hall↗