A canonical form for the continuous piecewise polynomial functions

TL;DR

This paper introduces a canonical form for continuous piecewise polynomial functions using a new class of functions, enabling easier representation, manipulation, and evaluation without relying on real algebraic numbers.

Contribution

The paper proposes a novel canonical form for continuous piecewise polynomial functions based on a specific class of functions, simplifying their manipulation and evaluation.

Findings

Provides a new representation for continuous piecewise polynomial functions.

Enables evaluation using only rational operations, avoiding real algebraic numbers.

Facilitates easier manipulation and analysis of such functions.

Abstract

We present in this paper a canonical form for the elements in the ring of continuous piecewise polynomial functions. This new representation is based on the use of a particular class of functions $$\{C_i(P):P\in\Q[x],i=0,\ldots,\deg(P)\}$$ defined by $$C_i(P)(x)= \left\{ \begin{array}{cll}0 & \mbox{ if } & x \leq \alpha \\ P(x) & \mbox{ if } & x \geq \alpha \end{array} \right.$$ where $\alpha$ is the $i$-th real root of the polynomial $P$. These functions will allow us to represent and manipulate easily every continuous piecewise polynomial function through the use of the corresponding canonical form. It will be also shown how to produce a "rational" representation of each function $C_{i}(P)$ allowing its evaluation by performing only operations in $\Q$ and avoiding the use of any real algebraic number.