What is BERNINA?

Piz Bernina is a 4049-meter peak in Graubünden, in the south-east corner of Switzerland. Its large summit, which appears on the cover of this guide, dominates the majestuous Morteratsch and Tschierva glaciers. This guide describes however BERNINA, an interactive interface to the $\Sigma{}^{{\rm it}}$ library that provides some efficient computations revolving around operators in $\mathbbm{Q}[x,\frac d{dx}]$ or $\mathbbm{Q}(x)[\frac d{dx}]$. In fact, the main goal of BERNINA is to provide access to selected functionalities of the $\Sigma{}^{{\rm it}}$ library to other computer algebra systems, or to users who do not own an Aldor compiler, or who feel more confortable with interactive access to a library. The functionalities currently provided by BERNINA are the ones that tend to be missing from several commercial computer algebra systems, namely:

• Basic arithmetic in $\mathbbm{Q}[x,\frac d{dx}]$, including adjoints and left and right gcd's and lcm's.
• Symmetric and exterior powers.
• Rational kernels, i.e.  $\mbox{Ker}(L) \cap \mathbbm{Q}(x)$ for $L \in \mathbbm{Q}[x,\frac d{dx}]$, and rational solutions of inhomogeneous equations of the form $L y = g$.
• Rational kernels of systems of the form $dY/dx = A(x) Y(x)$.
• Radical (resp. exponential) solutions, i.e. the solutions $y$ of $L y = 0$ such that $y^e \in \mathbbm{Q}(x)$ for some integer $e > 0$ (resp.  $y'/y \in \mathbbm{Q}(x)$).
• Liouvillian solutions of second-order operators.
• Factorisation and decomposition of second and third-order operators.

If you do not need any of the above computations, then you probably will not have much use for BERNINA. Otherwise, since BERNINA provides implementations of several recent algorithms for those computations, it is worth trying it and reading further.