An explicit non-elementary matrix over a bivariate Laurent polynomial ring

Abstract

Let $F$ be the fraction field of a discrete valuation ring. We give an explicit matrix in ${\mathrm{SL}}_2(F[X^{\pm 1},Y^{\pm 1}])\setminus {\mathrm{E}}_2(F[X^{\pm 1},Y^{\pm 1}])$. The proof of non-elementarity is based on techniques developed by Peter Abramenko in his preprint arXiv:0808.1095v1.

11 pages; comments are welcome!