1\chapter{CALI: Computational Commutative Algebra} 2\label{CALI} 3\typeout{{CALI: Computational Commutative Algebra}} 4 5{\footnotesize 6\begin{center} 7Hans-Gert Gr\"abe \\ 8Institut f\"ur Informatik, Universit\"at Leipzig\\ 9Augustusplatz 10 -- 11\\ 1004109 Leipzig, Germany \\[0.05in] 11e--mail: graebe@informatik.uni-leipzig.de 12\end{center} 13} 14 15\ttindex{CALI} 16 17This package contains algorithms for computations in commutative algebra 18closely related to the Gr\"obner algorithm for ideals and modules. Its 19heart is a new implementation of the Gr\"obner algorithm that also allows 20for the computation of syzygies. This implementation is also applicable to 21submodules of free modules with generators represented as rows of a matrix. 22As main topics CALI contains facilities for 23\begin{itemize} 24\item defining rings, ideals and modules, 25 26\item computing Gr\"obner bases and local standard bases, 27 28\item computing syzygies, resolutions and (graded) Betti numbers, 29 30\item computing (now also weighted) Hilbert series, multiplicities, 31independent sets, and dimensions, 32 33\item computing normal forms and representations, 34 35\item computing sums, products, intersections, quotients, stable 36quotients, elimination ideals etc., 37 38\item primality tests, computation of radicals, unmixed radicals, 39equidimensional parts, primary decompositions etc. of ideals and 40modules, 41 42\item advanced applications of Gr\"obner bases (blowup, associated graded 43ring, analytic spread, symmetric algebra, monomial curves etc.), 44 45\item applications of linear algebra techniques to zero dimensional 46 ideals, as {\em e.g.\ }the FGLM change of term orders, border bases 47 and affine and projective ideals of sets of points, 48 49\item splitting polynomial systems of equations mixing factorisation and 50the Gr\"obner algorithm, triangular systems, and different versions of the 51extended Gr\"obner factoriser. 52 53\end{itemize} 54 55There is more extended documentation on this package elsewhere, which 56includes facilities for tracing and switches to control its behaviour. 57 58