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Abstract: This paper outlines some new algorithms for operations on multivariate polynomials. The point of view taken in the paper is that all polynomials considered are given by their values at known points, and that the degrees of the polynomials are known or can be deduced. The first nontrivial algorithm considered is division; then Bezout matrices, generalized companion matricies, GCD, and the solution of zero-dimensional polynomial systems are investigated. A new algorithm is given for such solution: neither Gr\"obner basis nor resultant-based, it shares some characteristics with resultant algorithms.
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