Text Compactor

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Text Compactor is a free online tool that helps you summarize text. Simply enter your text, and the tool will generate a summary for you. It's easy to use and available on the website.

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Which problems in symbolic computation can be solved reasonably fast? Given a problem, would it be possible to predict how much time is needed to solve it? The area of computational complexity was designed to provide answers to such questions. However, symbolic computation has a few particularities that call for extra or separate techniques.

One particularity is that the most prominent algorithms typically get implemented, so we can check theory against practice. A satisfactory complexity theory in our area should at least be able to explain or model running times that are observed in practice.

However, such a theory should be more than an ennobled tool for benchmarking. We also wish to gain insight into the intrinsic complexity of problems, understand how the complexities of different problems are related, and identify the most central computational tasks in a given area (these tasks typically require highly optimized implementations).

Traditionally, a distinction is made between the analysis of algorithms (how to analyze the complexity of a given algorithm) and complexity theory (fitting problems into complexity classes and proving lower bounds). This paper partially falls in the second category, since we will be interested in classes of algorithms and problems. Such a broader perspective is particularly useful if you want to implement a new computer algebra system, like Mathemagix in my case [77], or a library for such a system.

However, traditional complexity theory is not very adequate for computer algebra, due to the disproportionate focus on lower bounds and coarse complexity classes. Fortunately, as will be argued in Section 3, we already have many ingredients for developing a more meaningful complexity theory for our area. I found it non-trivial to organize these ingredients into a “grand theory”. In this survey, I will therefore focus on concrete examples that reflect the general spirit.

The first series of examples in Section 4 concerns basic operations like multiplication, division, gcds, etc. for various algebras. We will see how various basic complexities are related and how some of them can be derived using general techniques.

In Section 5, we will turn to the higher level problem of polynomial system solving. My interest in this problem is more recent and driven by complexity considerations. I will survey advantages and limitations of the main existing approaches and highlight a few recent results.

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