Proposal of Multivariate Polynomial Arithmetic in a Specified Width of High- or Low-Exponents

The truncated power-series is very useful in computer algebra, but we must control the cutoff degree carefully when we use the power-series in actual algorithms. In this paper, we propose a more useful multivariate polynomial arithmetic. The arithmetic reserves only the terms of high- or low-exponents in a user-specified width with respect to a user-specified variable, of a multivariate polynomial being applied sequentially by many different kinds of arithmetic operations some of which cause large intermediate expression growths and some of others cause large expression shrinkage. Of course, the exponents are determined by the conventional arithmetic hence changed as the computation proceeds, while the width (measured from either the highest or the lowest exponent) is fixed. We investigate this idea from practical point of view, and give detailed procedures for realizing this idea concretely.