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In this paper, we propose a new method for removing all the redundant inequalities generated by Fourier-Motzkin elimination. This method is based on an improved version of Balas' work and can also be used to remove all the redundant inequalities in the input system. Moreover, our method only uses arithmetic operations on matrices and avoids resorting to linear programming techniques. Algebraic complexity estimates and experimental results show that our method outperforms alternative approaches, in particular those based on linear programming and simplex algorithm.

Fast algorithms for integer and polynomial multiplication play an important role in scientific computing as well as in other disciplines. In 1971, Schönhage and Strassen designed an algorithm that improved the multiplication time for two integers of at most n bits to O(log n log log n). In 2007, Martin Fürer presented a new algorithm that runs in O n log n ⋅ 2 O(log * n) , where log * n is the iterated logarithm of n.We explain how we can put Fürer's ideas into practice for multiplying polynomials over a prime field Z pZ, for which p is a Generalized Fermat prime of the form p = r k + 1 where k is a power of 2 and r is of machine word size. When k is at least 8, we show that multiplication inside such a prime field can be efficiently implemented via Fast Fourier Transform (FFT). Taking advantage of Cooley-Tukey tensor formula and the fact that r is a 2k-th primitive root of unity in Z pZ, we obtain an efficient implementation of FFT over Z pZ. This implementation outperforms comparable implementations either using other encodings of Z pZ or other ways to perform multiplication in Z pZ.