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In this talk I will present three probems of interest to the computer
algebra community. The first is the problem of implementing modular
algorithms efficiently. Application of the Chinese remainder theorem
to solve the GCD and Grobner bases problems leads to a big loss of
efficiency because the data structure overhead overwhelm's the cost of
the modular arithmetic. The second problem is how to build a system so
that all the components interact well. I will take as an example a
problem of automatic differentation from astrophysics where the
function to be differentiated involves the solution of a non-linear
equation. Can the CAS differentiate commands like fsolve(f=0,x=a); in
a program? The third problem is a problem of trying to implement
generic algorithms, efficiently. I will take as an example a linear
p-adic Newton iteration. A generic version of this algorithm would
work over Z mod p^k and over F[x] mod x^k for example.
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