Solving commutative relaxations of word problems

We present an algebraic characterization of the standard commutative relaxation of the word problem in terms of a polynomial equality. We then consider a variant of the commutative word problem, referred to as the "zero-to-all reachability" problem. We show that this problem is equivalent to a finite number of commutative word problems, and we use this insight to derive necessary conditions for zero-to-all reachability. We conclude with a set of illustrative examples.