We present version 4.0 of the symbolic manipulation system FORM. The most
important new features are manipulation of rational polynomials and the
factorization of expressions. Many other new functions and commands are also
added; some of them are very general, while others are designed for building
specific high level packages, such as one for Groebner bases. New is also the
checkpoint facility, that allows for periodic backups during long calculations.
Lastly, FORM 4.0 has become available as open source under the GNU General
Public License version 3.Comment: 26 pages. Uses axodra
Optimizing the cost of evaluating a polynomial is a classic problem in
computer science. For polynomials in one variable, Horner's method provides a
scheme for producing a computationally efficient form. For multivariate
polynomials it is possible to generalize Horner's method, but this leaves
freedom in the order of the variables. Traditionally, greedy schemes like
most-occurring variable first are used. This simple textbook algorithm has
given remarkably efficient results. Finding better algorithms has proved
difficult. In trying to improve upon the greedy scheme we have implemented
Monte Carlo tree search, a recent search method from the field of artificial
intelligence. This results in better Horner schemes and reduces the cost of
evaluating polynomials, sometimes by factors up to two.Comment: 5 page
Abstract. After a computer chess program had defeated the human World Champion in 1997, many researchers turned their attention to the oriental game of Go. It turned out that the minimax approach, so successful in chess, did not work in Go. Instead, after some ten years of intensive research, a new method was developed: MCTS (Monte Carlo Tree Search), with promising results. MCTS works by averaging the results of random play-outs. At first glance it is quite surprising that MCTS works so well. However, deeper analysis revealed the reasons.The success of MCTS in Go caused researchers to apply the method to other domains. In this article we report on experiments with MCTS for finding improved orderings for multivariate Horner schemes, a basic method for evaluating polynomials. We report on initial results, and continue with an investigation into two parameters that guide the MCTS search. Horner's rule turns out to be a fruitful testbed for MCTS, allowing easy experimentation with its parameters. The results reported here provide insight into how and why MCTS works. It will be interesting to see if these insights can be transferred to other domains, for example, back to Go.
We confirm a conjecture about the construction of basis elements for the multiple zeta values (MZVs) at weight 27 and weight 28. Both show as expected one element that is twofold extended. This is done with some lengthy computer algebra calculations using TFORM to determine explicit bases for the MZVs at these weights.
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