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math.RA — Rings and Algebras

A Universal Noncommutative Splitting Algebra for Polynomials

Contributed by Darij Grinberg

Let $R$ be a commutative ring. We show that every homogeneous polynomial $f\in R[x_{1},\ldots,x_{m}]$ of degree $n$ splits into a product of $n$ homogeneous linear forms over a suitable noncommutative ring extension $S$ of $R$ (that is, over a noncommutative $R$ -algebra $S$ whose structure morphism $R\rightarrow S$ is injective). The algebra $S$ is universal for such factorizations and is free as an $R$ -module. The proof uses Bergman's Diamond Lemma: we define $S$ by generators and relations, and the relations form a terminating reduction system that has no ambiguities and thus is confluent. An analogous result is also shown for inhomogeneous polynomials (with inhomogeneous factors). This easily follows from the homogeneous case by homogenizing and then setting the homogenizing variable equal to $1$ .

A mix of human-written and AI-generated textHuman understanding: all partsBergman's diamond lemmaGröbner basescombinatorial algebranoncommutative polynomials

Sep 29, 2026PDF ↗View work →

hexagon:2609.00121v1

math.AC — Commutative Algebra

Polynomials over Symmetric Polynomials

Contributed by Darij Grinberg

Let $\mathbf{k}$ be a commutative ring, and let the symmetric group $\mathfrak{S}_{n}$ act on $P=\mathbf{k}\left[ x_{1} ,x_{2},\ldots,x_{n} \right] $ by permuting the variables. We prove four results that are known (at least in the case when $\mathbf{k}$ is a field) but not easily found in the literature. First, the coinvariant algebra (the quotient of $P$ by the ideal generated by the symmetric polynomials with constant term $0$ ) is a free $\mathbf{k}$ -module of rank $n!$ , with the residue classes of the Artin monomials as a basis. Second, $P$ is a free module of rank $n!$ over the ring $P^{\mathfrak{S}_{n}}$ of symmetric polynomials, again with the Artin monomials as a basis. Third, if $n!$ is invertible in $\mathbf{k}$ , the coinvariant algebra is the regular $\mathbf{k}\left[ \mathfrak{S}_{n} \right] $-module. Fourth, under the same hypothesis, $P$ is a free left $P^{\mathfrak{S}_{n}}\left[ \mathfrak{S}_{n} \right]$ -module of rank $1$ . The first result follows from an elementary normal-form lemma for monic polynomials with pairwise relatively prime leading monomials. The second is proved by lifting the Artin basis. For the third, we use orbit harmonics with a strongly discrete point orbit, and the fourth follows by equivariantly lifting a regular basis of the coinvariant algebra.

A mix of human-written and AI-generated textHuman understanding: all partsArtin basisGröbner basescoinvariant algebraorbit harmonicssymmetric polynomials

Sep 28, 2026PDF ↗View work →

hexagon:2609.00117v1

math.AC — Commutative Algebra

Splitting a Polynomial into Linear Factors after an Injective Ring Extension

Contributed by Darij Grinberg

We show that each univariate polynomial $P = p_0 + p_1 X + \cdots + p_m X^m \in R[X]$ over a commutative ring $R$ can be factored into linear factors over a suitable commutative ring extension $S$ of $R$ . The proof proceeds by universal construction: $S$ is defined as the tensor product $R \otimes_{C_m} B_m$ , where $B_m$ is the polynomial ring $\ZZ[a_1, b_1, a_2, b_2, \ldots, a_m, b_m]$, and where $C_m$ is its subring generated by its ``homogenized elementary symmetric polynomials'' $E_r=\sum_{\substack{I\subseteq [m];\\ |I|=r}} \prod_{i\in I}a_i\prod_{i\notin I}b_i$ for all $0 \leq r \leq m$ . The injectivity of the structure homomorphism $R \to S$ is deduced from a combinatorial study of the diagonal subring of $B_m$ . In the process, a homogeneous variant of the Garsia--Stanton basis is constructed, and some classical properties of symmetric polynomials are recovered.

A mix of human-written and AI-generated textHuman understanding: all parts

Sep 28, 2026PDF ↗View work →

math.AC — Commutative Algebra
math.AG — Algebraic Geometry
math.AP — Analysis of PDEs
math.AT — Algebraic Topology
math.CA — Classical Analysis and ODEs
math.CO — Combinatorics
math.CT — Category Theory
math.CV — Complex Variables
math.DG — Differential Geometry
math.DS — Dynamical Systems
math.FA — Functional Analysis
math.GM — General Mathematics
math.GN — General Topology
math.GR — Group Theory
math.GT — Geometric Topology
math.HO — History and Overview
math.IT — Information Theory
math.KT — K-Theory and Homology
math.LO — Logic
math.MG — Metric Geometry
math.NA — Numerical Analysis
math.NT — Number Theory
math.OA — Operator Algebras
math.OC — Optimization and Control
math.PR — Probability
math.QA — Quantum Algebra
math.RA — Rings and Algebras
math.RT — Representation Theory
math.SG — Symplectic Geometry
math.SP — Spectral Theory
math.ST — Statistics Theory
cs.CC — Computational Complexity
cs.CG — Computational Geometry
cs.DM — Discrete Mathematics
cs.DS — Data Structures and Algorithms
cs.FL — Formal Languages and Automata Theory
cs.IT — Information Theory
cs.LO — Logic in Computer Science
cs.MS — Mathematical Software
cs.NA — Numerical Analysis
cs.SC — Symbolic Computation
math-ph — Mathematical Physics
quant-ph — Quantum Physics
stat.TH — Statistics Theory