Multi-point evaluation in higher dimensions

Abstract

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 $$\mathcal C $$ -algebras of the form $$\mathcal A =\mathcal C [x_1, {\ldots }, x_n] / I$$ , where $$I$$ is generated by monomials of the form $$x_1^{i_1} {\cdots } x_n^{i_n}$$ ; one particularly important example is the algebra of truncated power series $$\mathcal C [x_1, {\ldots }, x_n] / (x_1, {\ldots }, x_n)^d$$ . 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 Schost (ISSAC’05, ACM, New York, NY, pp 293–300, 2005), we obtain fast multiplication algorithms for algebras $$\mathcal A $$ of the above form.