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Elad Paran

Publications and source records attributed to Elad Paran.

At least 19 recordsLinked to original sources
arXiv2026

On Bayart's unique factorization problem

Let $A$ be a ring such that the formal power series ring $A[[t]]$ is a unique factorization domain. Bayart's problem asks whether the formal power series ring in two variables $A[[x,t]]$ must also be a UFD. Equivalently, is every irreducible element $f\in A[[x,t]]$ prime? We prove that the answer is yes whenever the constant term of $f$ is an associate of $p^m$, for any prime $p \in A$ and $1\leq m\leq3$, and for $m=4$ if $A$ contains $\Q$. We also prove the following reduction: To prove a positive answer to Bayart's problem, it suffices to prove that $A[x][[t]]$ is a UFD whenever $A[[t]]$ is.

math.AC↗
arXiv2026

Unique Factorization of Power Series Rings over Q-UFDs of Dimension at Most 3

Let $R$ be a Noetherian unique factorization domain containing $\mathbb Q$, with $\dim R\leq3$. We prove that $R[[t]]$ is a unique factorization domain if and only if two conditions hold: every two-generated parameter ideal $q$ in every height-two localization satisfies $\overline{q^2}\subseteq q$, and every height-three localization has depth three. No excellence or completion hypothesis is imposed. The main local result is that $A[[t]]$ is not a UFD whenever $A$ is a local $\mathbb Q$-UFD of dimension at least three and depth two. We also prove that the UFD property of $R[[t]]$ passes to every coefficient localization in dimension at most three, give an all-dimensional criterion when the completions of the maximal localizations satisfy Serre's condition $(S_2)$, and obtain an arbitrary-characteristic criterion for excellent UFDs of dimension at most three.

math.AC↗
arXiv2026

Characterization of the centrally finite Amitsur-Small division rings

In 1978, Amitsur and Small asked whether, for every division ring $D$, maximal left ideals of $D[x_1,\ldots,x_n]$ contract to maximal left ideals in smaller polynomial subrings. Chapman and the author showed that the answer is negative in general and called $D$ an Amitsur-Small ring when this contraction property always holds. Earlier work showed that Hamilton's real quaternion algebra is Amitsur-Small, that division algebras of degree three are not Amitsur-Small, and that degree-two examples are restricted to a specific quaternionic form; subsequent work excluded cyclic division algebras of odd prime degree. We give a complete resolution in the centrally finite case. If $D$ has finite dimension over its center $F$, then $D$ is an Amitsur-Small ring if and only if either $D=F$, or $F$ is real closed and $D$ is the Hamilton quaternion algebra $(-1,-1)_F$. In every other noncommutative centrally finite case, failure already occurs in two variables: there is a maximal left ideal $M\subseteq D[x,y]$ such that $M\cap D[x]$ is not maximal in $D[x]$.

math.RA↗
arXiv2026

The Combinatorial Nullstellensatz, Chevalley-Warning Theorem and weak Finitesatz in skew polynomial rings

We study zeros of polynomials in the multivariate skew polynomial ring $D[x_1,\ldots,x_n; σ]$, where $σ$ is an automorphism of a division ring $D$. We prove a generalization of Noga Alon's celebrated Combinatorial Nullstellensatz for such polynomials. In the case where $D$ is a finite field, we prove skew analogues of the Chevalley--Warning theorem, Ax's Lemma, and the weak case of Terjanian's Finitesatz.

math.AC↗
arXiv2026

On Landweber`s unique factorization problem

We solve a long-standing open problem, posed by Landweber in 1974: Let $R = K[x_1, x_2, . . . ]$ be the ring of polynomials in countably many variables over a field $K$. Is the formal power series ring $R[[t]]$ a unique factorization domain? We prove that it is. The proof is based on a new general result in commutative algebra: If $R$ is a Krull domain, and $f \in R[[t]]$ is irreducible, then $f$ is irreducible modulo a finite power of $t$.

math.AC↗
arXiv2025

Noether's normalization in skew polynomial rings

We study Noether's normalization lemma for finitely generated algebras over a division algebra. In its classical form, the lemma states that if $I$ is a proper ideal of the ring $R=F[t_1,\ldots,t_n]$ of polynomials over a field $F$, then the quotient ring $R/I$ is a finite extension of a polynomial ring over $F$. We prove that the lemma holds when $R=D[t_1,\ldots,t_n]$ is the ring of polynomials in $n$ central variables over a division algebra $D$. We provide examples demonstrating that Noether's normalization may fail for the skew polynomial ring $D[t_1,\ldots,t_n;σ_1,\ldots,σ_n]$ with respect to commuting automorphisms $σ_1,\ldots,σ_n$ of $D$. We give a sufficient condition for $σ_1,\ldots,σ_n$ under which the normalization lemma holds for such ring. In the case where $D=F$ is a field, this sufficient condition is proved to be necessary.

math.RA↗
arXiv2025

Images of polynomial maps and the Ax-Grothendieck theorem over algebraically closed division rings

We study the images of polynomial maps over algebraically closed division rings. Our first result generalizes the classical Ax-Grothendieck theorem: We show that if $ f_1, \ldots, f_m $ are elements of the free associative algebra $ D\langle X_1, \ldots, X_m \rangle $ generated by $ m \geq 1 $ variables over an algebraically closed division ring $ D $ of finite dimension over its center $ F $, and if the induced map $ f = (f_1, \ldots, f_m) \colon D^m \to D^m $ is injective, then $ f $ must be surjective. With no condition on the dimension over the center, our second result is that $ p(D) = D $ if $ p $ is either an element in $ F\langle X_1, \ldots, X_m \rangle $ with zero constant term such that $ p(F) \neq \{0\} $, or a nonconstant polynomial in $F[x]$. Furthermore, we also establish some Waring type results. For instance, for any integer $ n > 1 $, we prove that every matrix in $ \mathrm{M}_n(D) $ can be expressed as a difference of pairs of multiplicative commutators of elements from $ p(\mathrm{M}_n(D)) $, provided again that $ D $ is finite-dimensional over $ F $.

math.RA↗
arXiv2025

On the geometry of zero sets of central quaternionic polynomials II

Following the work of the first and last authors [2], we further analyze the structure of a zero set of a left ideal in the ring of central polynomials over the quaternion algebra H. We describe the "algebraic hull" of a point in H^n and prove it is a product of spheres. Using this description we give a new proof to a conjecture of Gori, Sarfatti and Vlacci. We also show that the main result of [2] does not extend to general division algebras.

math.RA↗
arXiv2024

On a problem of Amitsur and Small

We construct an example of a division ring $D$ and a maximal left ideal $M$ in the polynomial ring $D[x,y]$ in two central variables over $D$, such that the intersection $M \cap D[x]$ is not a maximal left ideal in $D[x]$. This resolves a ring-theoretic problem of Amitsur and Small raised in 1978.

math.AC↗
arXiv2024

The Collatz map analogue in polynomial rings and in completions

We study an analogue of the Collatz map in the polynomial ring $R[x]$, where $R$ is an arbitrary commutative ring. We prove that if $R$ is of positive characteristic, then every polynomial in $R[x]$ is eventually periodic with respect to this map. This extends previous works of the authors and of Hicks, Mullen, Yucas and Zavislak, who studied the Collatz map on $\mathbb{F}_p[x]$ and $\mathbb{F}_2[x]$, respectively. We also consider the Collatz map on the ring of formal power series $R[[x]]$ when $R$ is finite: we characterize the eventually periodic series in this ring, and give formulas for the number of cycles induced by the Collatz map, of any given length. We provide similar formulas for the original Collatz map defined on the ring $\mathbb{Z}_2$ of $2$-adic integers, extending previous results of Lagarias.

math.CO↗
arXiv2024

On the geometry of zero sets of central quaternionic polynomials

Let R be the ring of polynomials in n central variables over the real quaternion algebra H, and let I be a left ideal in R. We prove that if a polynomial p in R vanishes at all the common zeros of I in H^n with commuting coordinates, then as a slice regular quaternionic function, p vanishes at all common zeros of I in H^n. This confirms a conjecture of Gori, Sarfatti and Vlacci, who settled the two dimensional case.

math.RA↗
arXiv2024

On the stopping time of the Collatz map in $\mathbb{F}_2[x]$

We study the stopping time of the Collatz map for a polynomial $f \in \mathbb{F}_2[x]$, and bound it by $O({\rm deg} (f)^{1.5})$, improving upon the quadratic bound proven by Hicks, Mullen, Yucas and Zavislak. We also prove the existence arithmetic sequences of unbounded length in the stopping times of certain sequences of polynomials, a phenomenon observed in the classical Collatz map.

math.CO↗
arXiv2023

A skew Newton-Puiseux Theorem

We prove a skew generalization of the Newton-Puiseux theorem for the field $F = \bigcup_{n=1}^\infty \mathbb{C}((x^\frac{1}{n}))$ of Puiseux series: For any positive real number $α$, we consider the $\mathbb{C}$-automorphism $σ$ of $F$ given by $x \mapsto αx$, and prove that every non-constant polynomial in the skew polynomial ring $F[t,σ]$ factors into a product of linear terms. This generalizes the classical theorem where $σ= {\rm id}$, and gives the first concrete example of a field of characteristic $0$ that is algebraically closed with respect to a non-trivial automorphism -- a notion studied in works of Aryapoor and of Smith. Our result also resolves an open question of Aryapoor concerning such fields. A key ingredient in the proof is a new variant of Hensel's lemma.

math.RA↗
arXiv2023

Lüroth's and Igusa's theorems over Division Rings

Let $H$ be a division ring of finite dimension over its center, let $H[T]$ be the ring of polynomials in a central variable over $H$, and let $H(T)$ be its quotient skew field. We show that every intermediate division ring between $H$ and $H(T)$ is itself of the form $H(f)$, for some $f$ in the center of $H(T)$. This generalizes the classical Lüroth's theorem. More generally, we extend Igusa's theorem characterizing the transcendence degree 1 subfields of rational function fields, from fields to division rings.

math.NT↗
arXiv2022

Fixed points and orbits in skew polynomial rings

We study orbits and fixed points of polynomials in a general skew polynomial ring $D[x,σ, δ]$. We extend results of the first author and Vishkautsan on polynomial dynamics in $D[x]$. In particular, we show that if $a \in D$ and $f \in D[x,σ,δ]$ satisfy $f(a) = a$, then $f^{\circ n}(a) = a$ for every formal power of $f$. More generally, we give a sufficient condition for a point $a$ to be $r$-periodic with respect to a polynomial $f$. Our proofs build upon foundational results on skew polynomial rings due to Lam and Leroy.

math.RA↗
arXiv2020

A Quaternionic Nullstellensatz

We prove a Nullstellensatz for the ring of polynomial functions in n non-commuting variables over Hamilton's ring of real quaternions. We also characterize the generalized polynomial identities in n variables which hold over the quaternions, and more generally, over any division algebra.

math.RA↗
arXiv2018

Embedding problems for automorphism groups of field extensions

A central conjecture in inverse Galois theory, proposed by Dèbes and Deschamps, asserts that every finite split embedding problem over an arbitrary field can be regularly solved. We give an unconditional proof of a consequence of this conjecture, namely that such embedding problems can be regularly solved if one waives the requirement that the solution fields are normal. This extends previous results of M. Fried, Takahashi, Deschamps, and the last two authors concerning the realization of finite groups as automorphism groups of field extensions.

math.NT↗