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Stefan Barańczuk

Publications and source records attributed to Stefan Barańczuk.

9 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↗
arXiv2025

Divisibility sequences related to abelian varieties isogenous to a power of an elliptic curve

Let $A$ be an abelian variety defined over a number field $K$, $E/K$ be an elliptic curve, and $ϕ:A\to E^m$ be an isogeny defined over $K$. Let $P\in A(K)$ be such that $ϕ(P)=(Q_1,\dots, Q_m)$ with $\text{Rank}_\mathbb{Z}(\langle Q_1,\dots, Q_m\rangle)=1$. We will study a divisibility sequence related to the point $P$ and show its relation with elliptic divisibility sequences.

math.NT↗
arXiv2024

Hasse-Minkowski theorem for quadratic forms on groups

Consider groups such as Mordell-Weil groups of abelian varieties over number fields, odd algebraic $K$-theory groups of number fields, or finitely generated subgroups of the multiplicative groups of number fields. They are all equipped with systems of reduction maps; thus, one can investigate the Hasse-Minkowski theorem for quadratic forms with coefficients in such groups. In this paper, we prove that the theorem holds for the forms whose rank equals $2$ or $3$, and we demonstrate that it does not hold for higher ranks by providing a counterexample. We also show that our results constitute a generalization of the classic Hasse-Minkowski theorem for binary and ternary integral forms.

math.NT↗
arXiv2022

Reducing the number of equations defining a subset of the $n$-space over a finite field

Let $f_{1}, \ldots, f_{k}$ be polynomials defining an algebraic set in affine $n$-space over a finite field. Suppose $k>n$. We prove that there exists a system of polynomials $g_{1}, \ldots, g_{n}$, each being a linear combination with scalar coefficients of $f_{1}, \ldots, f_{k}$, defining the same algebraic set. In particular, one reduces the number of equations without increasing the total degree. We also have the corresponding result for systems of homogeneous polynomials defining algebraic sets in projective spaces.

math.AG↗
arXiv2021

Zsigmondy's Theorem for Chebyshev polynomials

For an integer $a$ consider the sequence $(T_{n}(a)-1)_{n=1}^{\infty}$ defined by the Chebyshev polynomials $T_{n}$. We list all pairs $(n,a)$ for which the term $T_{n}(a)-1$ has no primitive prime divisor.

math.NT↗