Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 0707.1552

Polynomials with a common composite

Robert M. Beals·Joseph L. Wetherell·Michael E. Zieve
View original source ↗Read source document ↗Export metadata ↓

Abstract

Let f and g be nonconstant polynomials over an arbitrary field K. In this paper we study the intersection of the polynomial rings K[f] and K[g], and in particular we ask whether this intersection is larger than K. We completely resolve this question when K has characteristic zero, and in positive characteristic we present various results, examples, and algorithms.

Explore related subjects

math.AGmath.NT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Robert M. Beals, Joseph L. Wetherell, Michael E. Zieve. 2008-04-07. Polynomials with a common composite. https://arxiv.org/abs/0707.1552

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

G3-Criteria and Applications

The G3-property of a subvariety was introduced by Hironaka-Matsumura, and plays an important role for deducing connectedness and extension results. Unfortunately, it's a rather elusive notion, which is not always easy to establish. Most of the existing work is concentrated on subvarieties of homogeneous varieties. The first goal of this article is to show that mobility assumptions on the subvariety, considered in works of Badescu, Chow, Debarre, Voisin, yield a certain partial positivity property, slightly stronger than G3, previously introduced by the author. Second, we apply the result to prove that, in numerous situations, the splitting of the normal bundle of a smooth two-codimensional subvariety implies that it is a complete intersection.

math.AG↗
arXiv2026

Moduli Stacks of $G$-Curves in Homotopy Theory at Height $p-1$

Let $p$ be odd and $G' = \mathbb{Z}/p \rtimes \mathbb{Z}/(p-1)^2$ the maximal finite subgroup of the Morava stabilizer group at height $p-1$. Inverse Galois theory produces from $G'$ alone a curve $X$, the unique curve of minimal genus with $\operatorname{Aut}(X) \simeq G'$; its ramification, its field of definition and its equation are consequences of the group, not choices. We prove a $G'$-equivariant equivalence between the deformations of $X$ and Lubin--Tate space, so that the Lubin--Tate action of $G'$ is the action of $\operatorname{Aut}(X)$ on deformations of the curve. The proof is a coordinate-free Kodaira--Spencer argument reducing to a single character count. The action becomes explicit: $G'$ acts through $\mathbb{F}_p \rtimes \mathbb{F}_p^\times$ shifting and scaling $p+1$ points on $\mathbb{P}^1$. From this we compute $H^*(G', π_* E_{p-1})$ and its Tate cohomology. One identity, $π^{p-1} = -p$, runs through every section.

math.AG↗