On a Rank Nullstellensatz Conjecture for Noncommutative Polynomials

Abstract

Jurij Vol\v{c}i\v{c} conjectured that a noncommutative polynomial $g$ belongs to the two-sided ideal generated by $f_1,\ldots,f_l$ if and only if, for matrices of every size, the rank of the evaluation of $g$ is bounded by a constant times the maximum rank of the evaluations of $f_1,\ldots,f_l$. In this paper, we show this equivalence when the generators $f_1,\ldots,f_l$ are homogeneous. We then give an explicit counterexample in the non-homogeneous case.