Starting with Newton’s laws, most physical theories deal with smoothly changing quantities. Differential equations describe how these quantities change with time. There exist efficient numerical methods for the accurate solution of such equations. The main idea behind these methods is that a smooth process can be approximated, with arbitrary accuracy, by piecewise linear (or piecewise polynomial) functions, and that we can determine the parameters for these approximations, for example, if we replace derivatives ∂f/∂t by finite differences . It is also known that analytical and algebraic transformations can drastically enhance the efficiency and accuracy of numerical methods.

In real life, in addition to smooth processes, we often observe abrupt (discontinuous, nonsmooth) changes like phase transitions, cracks, and so on. In macrophysics, in principle, such singularities can be viewed as approximations to the actually smooth (but fast) process; however, in fundamental physics, like general relativity or quantum field theory, singularities are real. Nonsmooth processes cannot be approximated by piecewise linear ones, and, thus, such processes are very difficult to analyze numerically. This difficulty cannot be resolved by using other types of approximations: indeed, for smooth processes, it is sufficient to know an approximate dynamic, and we can still get good predictions; in contrast, for singular processes, we need to know the exact form of the singularity to make accurate predictions.

What type of singularity do we encounter in such situations? Equations of fundamental physics can be presented in the equivalent form S → min, where the action S has the form S = ∫ L, dV → min for some function L polynomially depending on the values of the physical fields and their derivatives. Thus, the corresponding variational equations are algebraic (one or several polynomials P_j(x₁, . . . , x_m) of the unknown values x_i are equal to 0). So, the singularities of interest are also algebraic.

In most physical applications, like phase transitions, we encounter the simplest forms of algebraic singularities, such as cusp (corresponding to a simple transition) and swallowtail (corresponding to a triple point), which have been popularized by the catastrophe theory. In many situations, however, we have more complex singularities. In principle, researchers in algebraic geometry and commutative algebra have found out how to analyze such singularities, but the corresponding algorithms are very tedious, and require a computer to handle situations with many variables. Intuition does not always help much here: suffice it to say that Edward Witten, from Princeton’s Institute of Advances Studies, the world’s leading authority in superstring theory, whose intuition has enhanced numerous areas of mathematics and physics, had several singularity-related conjectures that turned out to be false.

The existing general-purpose computer algebra packages are also not very helpful. To handle singularities, a special computer algebra system called Singular was launched in 1982 by Gert-Martin Greuel, Gerhard Pfister, and others. This system has now grown into a major tool for singularity-related problems.

This edited book, dedicated to Greuel on his sixtieth birthday, contains chapters on the corresponding singularity-related problems, on the current state of Singular, and on the numerous applications of Singular to different problems. This book is mainly aimed at mathematicians who are familiar with the main ideas and techniques of singularity theory; it only gives a sketchy description of its computer implementation, so, by itself, it is not easily accessible for computer scientists. However, its chapters have good bibliographies, and, thus, the book provides a unique introduction to a newly developed area of singularity-related computations, an introduction highly recommended to anyone interested in this fast-growing area.