Uniform two-generator presentations for $S{L}_n(\mathbb{Z})$ with polynomial complexity bounds

We give a uniform explicit construction of finite two-generator presentations for the special linear groups over the integers in all ranks at least three. The construction builds on the generating-pair work of Conder–Liversidge–Vsemirnov and on a standard Tietze-elimination observation pointed out by Button. It recovers Trott’s odd-rank generating pair and extends the same monomial/transvection form uniformly to even rank by a sign correction. After rebalancing, the construction has quadratic transvection words, quartically many relators, and sextic total relator length. We also derive several consequences, including infinite–infinite and finite–finite variants, consequences for congruence quotients, a presentation for the projective quotient, and an exact relator count, valid for both the unbalanced and balanced presentations.