Building Bricks with Bricks, with Mathematica
Pietro Codara, Ottavio M. D'Antona, Daniele Filaretti

TL;DR
This paper presents an algorithm in Mathematica for constructing n-dimensional parallelepipeds based on multivariate polynomial bases, extending previous work on univariate polynomial decomposition to multivariate cases with visual step-by-step demonstrations for dimensions 2 to 4.
Contribution
It introduces a method to build multivariate parallelepipeds from polynomial bases, generalizing prior univariate approaches, with explicit algorithms and visualizations for low dimensions.
Findings
Algorithm successfully constructs parallelepipeds for multivariate polynomials.
Mathematica visualizations demonstrate step-by-step construction for dimensions 2 to 4.
Extension of polynomial basis decomposition to multivariate geometric constructions.
Abstract
In this work we solve a special case of the problem of building an n-dimensional parallelepiped using a given set of n-dimensional parallelepipeds. Consider the identity x^3 = x(x-1)(x-2)+3x(x-1+x). For sufficiently large x, we associate with x^3 a cube with edges of size x, with x(x-1)(x-2) a parallelepiped with edges x, x-1, x-2, with 3x(x-1+x) three parallelepipeds of edges x, x-1, 1, and with x a parallelepiped of edges x, 1, 1. The problem we takle is the actual construction of the cube using the given parallelepipeds. In [DDNP90] it was shown how to solve this specific problem and all similar instances in which a (monic) polynomial is expressed as a linear combination of a persistent basis. That is to say a sequence of polynomials q_0 = 1, and q_k(x) = q_{k-1}(x)(x-r_k) for k > 0. Here, after [Fil10], we deal with a multivariate version of the problem with respect to a basis of…
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Taxonomy
TopicsMathematics and Applications · Polynomial and algebraic computation · Advanced Combinatorial Mathematics
