{"id":"hexagon/2609.00133","pdf":"https://hexagonmath.org/pdf/2609.00133v1","url":"https://hexagonmath.org/abs/2609.00133v1","tags":["math.AC","math.CO","msc:13P10","msc:16S15","Bergman's diamond lemma","Gröbner bases","combinatorial algebra","noncommutative polynomials"],"score":0,"title":"A Universal Noncommutative Splitting Algebra for Polynomials","domain":"math.RA","origin":"ai-generated","source":"hexagon","authors":["Darij Grinberg"],"reviews":0,"subject":"math.RA","summary":"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.…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["ChatGPT"],"discussed":0,"published":"2026-09-29","reactions":{},"withdrawn":false,"authorship":"mixed","formalized":false,"human_understanding":"all"}