## Conclusions

• OBSERVATIONS.
• Computer algebra algorithms perform symbolic computations: numbers are manipulated by using their mathematical definitions.
• Hence results are exact and complete.
• However they can be huge!
• Moreover intermediate expressions may be much bigger than the input and output.
• OBJECTIVES.
• Our main interest will be here to study the implementation of algorithms
• that keep the swell of the intermediate expressions under control
• and offer optimal time complexity.
• Sine this is obviously a vast subject we will focus on univariate polynomials, bivariate polynomials and matrix operations.
• But in fact these algorithms benefit to many other Computer algebra algorithms.
• TYPICAL (TRIVIAL) EXAMPLE.
• Let and be two univariate polynomials over the integer numbers and with degrees and respectively such that .
• Suppose we want to decide whether divides .
• One can show that the division with remainder of by requires at most operations in the coefficient ring.
• If divides then for a given integer value of the integer divides .
• Computing and require operations in the coefficient ring.
• So trying whether divides (before computing the remainder of by ) will save time (and space) in the average.

Marc Moreno Maza
2008-01-07