This article takes an analytical viewpoint to address the following questions: 1. How can we justifiably beautify an input or result sum of non-numeric terms that has some approximate coefficients by deleting some terms and/or rounding some coefficients to simpler floating-point or rational numbers? 2. When we add two expressions, how can we justifiably delete more non-zero result terms and/or round some result coefficients to even simpler floating-point, rational or irrational numbers?The methods considered in this paper provide a justifiable scale-invariant way to attack these problems for subexpressions that are multivariate sums of monomials with real exponents.
Let K be a field and
x
1
< ··· <
x
n
be ordered variables. Consider a set of polynomials
F
⊂ K[
x
1
,...,
x
n
]. If the zero set
V
(
F
) of
F
is of positive dimension, then any triangular decomposition
One of the core commands in the RegularChains library isTriangularize. The underlying decomposes the solution set of anpolynomial system into geometrically meaningful components representedby regular chains. This algorithm works by repeatedly calling aprocedure, called Intersect, which computes the common zeros of apolynomial p and a regular chain T.As the number of variables of p and T, as well as their degrees,increase, the call Intersect(p, T) becomes more and morecomputationally expensive. It was observed in (C. Chen an M. MorenoMaza, JSC 2012) that when the input polynomial system iszero-dimensional and T is one-dimensional then this cost can besubstantially reduced. The method proposed by the authors is aprobabilistic algorithm based on evaluation and interpolationtechniques. This is the type of method which is typically challengingto implement in a high-level language like Maple's language, as asharp control of computing resources (in particular memory) is needed.In this paper, we report on a successful Maple implementation of thisalgorithm. We take advantage of Maple's modp1 function which offersfast arithmetic for univariate polynomials over a prime field.The method avoids unlucky specialization and the probabilistic aspectonly comes from the fact that non-generic solutions are notcomputed.
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