## 2 Polynomial Solvability
### General Case  
At any \(N\) we may study the QES sets of coupled nonlinear algebraic equations using the computer-assisted elimination technique of Grobner bases. In the even-parity case this algorithm generates certain polynomials \(P^{(N)}(d)\) which specify the QES-compatible shifts \(d\) as their zeros (see Table 1). The odd-parity QES solutions appear determined, similarly, via the values of \(a=a_{2}=-d\) which coincide with the real zeros of other polynomials \(Q^{(N)}(a)\) (see Table 2). The most compact representation of the above set of results may be obtained when one imagines that the even- and/or odd-parity boundary condition (14) at \(x=0\) may be reinterpreted as the respective initial, additional (i.e., \(n=-1\)) condition for recurrences (15). This enables us to solve these recurrences in terms of compact formulae. **Theorem 1**. : _In the even-parity QES case the general closed-form solution of recurrences (15) reads_  
\[a_{k}=a_{k}^{(N)}=\frac{(-1)^{k}}{k!(k-1)!}\det\mathcal{P}^{(N,k)}(d)\,,\;\; \;k=(1,2,)\,3,4,\ldots\,. \tag{19}\] \begin{table}
\begin{tabular}{c c c} \hline \(N\) & **Polynomial \(\mathcal{Q}^{(N)}\left(a\right)\)** & **Elimination of \(a_{N}\)** \\ \hline
2 & \(-5+2\,d^{2}\) & \(-1+2\,a_{2}=0\) \\
3 & \(3-9\,d^{2}+2\,d^{4}\) & \(-8\,d+2\,d^{3}+3\,a_{3}=0\) \\
4 & \(27-28\,d^{2}+4\,d^{4}\) & \(11-2\,d^{2}+12\,a_{4}=0\) \\
5 & \(-15+75\,d^{2}-40\,d^{4}+4\,d^{6}\) & \(29\,d-19\,d^{3}+2\,d^{5}+15\,a_{5}=0\) \\ � & \(\ldots\) & \(\ldots\) \\ \hline \end{tabular}
\end{table} 
Table 1: QES even-parity values of shifts \(d\) are defined, implicitly, as zeros of polynomials \(P^{(N)}(d)\). A sample of the Gröbnerian elimination is added. \begin{table}
\begin{tabular}{c c c} \hline \(N\) & **Polynomial \(\mathcal{P}^{(N)}\left(d\right)\)** & **Elimination of \(a_{N}\)** \\ \hline
2 & \(-5+2\,d^{2}\) & \(-1+2\,a_{2}=0\) \\
3 & \(3-9\,d^{2}+2\,d^{4}\) & \(-8\,d+2\,d^{3}+3\,a_{3}=0\) \\
4 & \(27-28\,d^{2}+4\,d^{4}\) & \(11-2\,d^{2}+12\,a_{4}=0\) \\
5 & \(-15+75\,d^{2}-40\,d^{4}+4\,d^{6}\) & \(29\,d-19\,d^{3}+2\,d^{5}+15\,a_{5}=0\) \\ � & \(\ldots\) & \(\ldots\) \\ \hline \end{tabular}
\end{table} 
Table 2: QES values \(a=a_{2}=-d\) determined as zeros of polynomials \(Q^{(N)}(a)\) (odd-parity case). _Its d-dependence is encoded in the tridiagonal \(k\) by \(k\) matrices_  
\[\mathcal{P}^{(N,k)}(d)=\left( \begin{array}{cccccc}d&1&0&0&\ldots&0\\ 2N&2d&2&0&\ddots&\vdots\\ 0&2N-2&4d&6&\ddots&0\\ 0&\ddots&\ddots&\ddots&\ddots&0\\ \vdots&\ddots&0&2(N-k+3)&2(k-2)d&(k-1)(k-2)\\ 0&\ldots&0&0&2(N-k+2)&2(k-1)d\end{array} \right). \tag{20}\]  
_These formulae must be complemented by the constraint \(a_{N+1}^{(N)}=0\), i.e., by the polynomial algebraic equation_  
\[\det\mathcal{P}^{(N,N+1)}(d)=0 \tag{21}\]  
_which represents an implicit definition of all of the admissible values of the specific, QES-compatible shift parameters \(d=d^{(QES)}\)._  
Proof. : One may proceed by mathematical induction. It is only necessary to keep in mind that in matrix (20) the first row reflecting the initial condition looks (and is) anomalous. That is why the form and validity of the \(k=1\) solution \(a_{1}=-d\) and also of the \(k=2\) solution \(a_{2}=d^{2}-N\) are to be verified separately. **Theorem 2**. : _In the odd-parity QES case the closed-form solution of recurrences (15) reads_  
\[a_{k+1}=a_{k+1}^{(N)}=\frac{(-1)^{k}}{(k+1)!k!}\det\mathcal{Q}^{(N,k)}(d)\,, \quad k=1,2,\ldots \tag{22}\]  
_with the d-dependence encoded in the tridiagonal \(k\) by \(k\) matrices_  
\[\mathcal{Q}^{(N,k)}(d)=\left( \begin{array}{cccccc}2d&2&0&\ldots&0\\ 2N-2&4d&6&\ddots&\vdots\\ 0&2N-4&\ddots&\ddots&0\\ \vdots&\ddots&\ddots&2(k-1)d&k(k-1)\\ 0&\ldots&0&2(N-k+1)&2 kd\end{array} \right). \tag{23}\]  
_This must be complemented by the algebraic polynomial-equation definition \(a_{N+1}^{(N)}=0\), i.e.,_  
\[\det\mathcal{Q}^{(N,N)}(d)=0 \tag{24}\]  
_yielding all of the admissible values of the specific, QES-compatible shift parameters \(d=d^{(QES)}\)._  
Proof.