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In this paper, we propose efficient new algorithms for multi-dimensional multi-point evaluation and interpolation on certain subsets of so called tensor product grids. These point-sets naturally occur in the design of efficient multiplication algorithms for finite-dimensional -algebras of the form , where is generated by monomials of the form ; one particularly important example is the algebra of truncated power series . Similarly to what is known for multi-point evaluation and interpolation in the univariate case, our algorithms have quasi-linear time complexity. As a known consequence, we obtain fast multiplication algorithms for algebras of the above form.

**Authors:**

**Keywords: **multi-point evaluation, multi-point
interpolation, algorithm, complexity, power series multiplication

**A.M.S. subject classification: **12Y05, 68W30, 68W40,
13P10, 65F99