Productive polynomials (Q1318188)

Let \(\mathbb{F}\) be a field and let \(\mathbb{F}\langle x_ 1,\dots, x_ n\rangle\) be the linear algebra of polynomials in the non-commuting indeterminates \(x_ 1,\dots, x_ n\). Let \(q\in \mathbb{F}\langle x_ 1,\dots, x_ n\rangle\). Let \(A\) be an associative algebra over \(\mathbb{F}\). \(q\) defines a mapping \(\widehat{q}\) of \(A\times\dots \times A= A^ n\) into \(A\) whose value \(\widehat{q}(a_ 1,\dots, a_ n)\) at \((a_ 1,\dots, a_ n)\) is the result of replacing each \(x_ i\) in \(q\) by the corresponding \(a_ i\), and then carrying out the algebraic operations proper to \(A\). A linear subspace \(B\) of the algebra \(A\) will be called \(q\)-closed if whenever \({\mathbf a}= (a_ 1,\dots, a_ n)\in A^ n\) then \(\widehat{q}({\mathbf a})\in B\). Let \(q((B))\) be the smallest \(q\)-closed linear subspace containing \(B\). We study mainly the case that \(A\) is \(\mathbb{F} \langle x_ 1,\dots, x_ n\rangle\) itself, and \(B\) is the linear subspace generated by \(x_ 1,\dots, x_ n\) and the unit 1. The \(q\)- closed set generated by \(x_ 1,\dots, x_ n\) and 1 will be denoted in this case simply by \(((q))\). We use \(P\) to stand for \(\mathbb{F}\langle x_ 1,\dots, x_ n\rangle\). \(q\in P\) will be called productive if \(((q))=P\); and otherwise, non-productive. Two questions are answered by us. 1. When is a given \(q\in P\) productive, and 2. if it is not, how to find elements \(p\) which are not in \(((q))\)?