A polynomial criterion for Jordan elements in a free associative algebra
Let $A=\Phi\langle X\rangle$ be a free associative algebra over a field of characteristic zero, and let $J$ be the Jordan subalgebra of $A^{(+)}$ generated by $X$ and $1$. For every $n\geq1$ we construct an element $U_n\in\mathbb Q[\mathfrak S_n]$ whose image on $A_n$ is exactly $J_n$. If \[ \det(tI-U_n|_{V_n})=t^{e_n}q_n(t),\qquad q_n(0)\ne0, \] on the multilinear component $V_n$, then \[ a\in J_n\quad\Longleftrightarrow\quad a\,q_n(U_n)=0. \] Thus the criterion gives a finite algorithm for recognizing Jordan elements: in each degree one constructs $U_n$ and $q_n$ and tests the single equation $a\,q_n(U_n)=0$. Moreover, $I-q_n(U_n)/q_n(0)$ is a projection of $A_n$ onto $J_n$. We give the projection explicitly in degrees at most four, record the multilinear dimensions through degree eight, and formulate the analogous criterion on each fixed multihomogeneous component.