Defining Polynomials Through Sequences: A Comprehensive Guide

Polynomials are often introduced as expressions involving variables and coefficients, but their formal definition can be surprisingly complex. Have you ever wondered if polynomials can be defined by sequences? This article aims to clarify this concept by exploring how polynomials can indeed be represented as sequences of coefficients, and delving into the implications of such a definition in the context of abstract algebra.

Polynomials and Sequences: A Formal Definition

Polynomials of one variable can be formally defined by sequences of coefficients. This is particularly useful when the variable is suppressed, allowing us to treat polynomials in a more abstract manner. Consider the polynomial:

4 - 2x^2 - 5x^3

This polynomial can be represented as the sequence:

4, 0, -2, -5, 0, 0, ...

where all the terms after -5 are zero. This sequence approach simplifies the representation of polynomials and facilitates operations like addition and subtraction, which can be performed coordinatewise. For example, adding the polynomials 4 - 2x^2 - 5x^3 and 1 3x - x^3 gives the sequence:

5, 3, -3, -6, 0, 0, ...

Multiplication of Polynomials via Sequences

Multiplication of polynomials defined as sequences is more complex. The product of two sequences, each representing a polynomial, is given by:

For sequences a (a_0, a_1, a_2, ...) and b (b_0, b_1, b_2, ...), their product is defined as:

(a_0, a_1, a_2, ...) * (b_0, b_1, b_2, ...) (a_0b_0, a_0b_1 a_1b_0, a_0b_2 a_1b_1 a_2b_0, ...),

where the nth term of the product is given by:

sum_{k0}^n a_kb_{n-k}

This definition of polynomials through sequences is particularly advantageous in abstract algebra, as it allows polynomials to be defined purely in terms of set theory without the need to specify the nature of the variables involved.

Implications in Abstract Algebra

The definition of polynomials through sequences also sheds light on a fundamental concept in abstract algebra: polynomials are not always the same as polynomial functions. When working with polynomials over an arbitrary field or ring, especially when the characteristic of the field is a prime number p rather than 0, different polynomials can sometimes produce the same function.

For example, in the field of integers modulo p, the polynomial xp is indistinguishable from the constant polynomial c for any constant c. This is because xp x in this context. Therefore, in such fields, many polynomial functions can be represented by different polynomials, leading to interesting theorems and insights.

To illustrate, the polynomial xp can be represented as the sequence:

0, 0, ..., 1, 0, 0, ... (1 in the pth slot, or 0th in natural indexing)

whereas the polynomial x is represented as:

0, 1, 0, 0, ...

This distinction is crucial in abstract algebra, as it highlights the importance of understanding the underlying field or ring when working with polynomials.