Polynomial arithmetic
Polynomial arithmetic is a branch of algebra dealing with some properties of polynomials which share strong analogies with properties of number theory relative to integers. It includes basic mathematical operations such as addition, subtraction, and multiplication, as well as more elaborate operations like Euclidean division, and properties related to roots of polynomials. The latter are essentially connected to the fact that the set K[X] of univariate polynomials with coefficients in a field K is a commutative ring, such as the ring of integers [math]\displaystyle{ \mathbb{Z} }[/math].
Elementary operations on polynomials
Addition and subtraction of two polynomials are performed by adding or subtracting corresponding coefficients. If
- [math]\displaystyle{ f(x) = \sum_{i=0}^n a_ix^i; g(x) = \sum_{i=0}^m b_ix^i }[/math]
then addition is defined as
- [math]\displaystyle{ f(x)+g(x)= \sum_{i=0}^m (a_i+b_i)x^i }[/math] where m > n
Multiplication is performed much the same way as addition and subtraction, but instead by multiplying the corresponding coefficients. If [math]\displaystyle{ f(x) = \sum_{i=0}^n a_ix^i; g(x) = \sum_{i=0}^m b_ix^i }[/math] then multiplication is defined as [math]\displaystyle{ f(x)\times g(x)=\sum_{i=0}^{n+m} c_ix^i }[/math] where [math]\displaystyle{ c_k=a_0b_k+a_1b_{k-1}+\cdots+a_{k-1}b_1+a_kb_0 }[/math]. Note that we treat [math]\displaystyle{ a_i }[/math] as zero for [math]\displaystyle{ i\gt n }[/math] and that the degree of the product is equal to the sum of the degrees of the two polynomials.
Advanced polynomial arithmetics and comparison with number theory
Many fascinating properties of polynomials can be found when, thanks to the basic operations that can be performed on two polynomials and the underlying commutative ring structure of the set they live in, one tries to apply reasonings similar to those known from number theory.
To see this, one first needs to introduce two concepts: the notion of root of a polynomial and that of divisibility for pairs of polynomials.
If one considers a polynomial [math]\displaystyle{ P }[/math] of a single variable X in a field K (typically [math]\displaystyle{ \mathbb{R} }[/math] or [math]\displaystyle{ \mathbb{C} }[/math]), and with coefficients in that field, a root [math]\displaystyle{ r }[/math] of [math]\displaystyle{ P }[/math] is an element of K such that
- [math]\displaystyle{ P(r)=0 }[/math]
The second concept, divisibility of polynomials, allows to see a first analogy with number theory: a polynomial [math]\displaystyle{ B }[/math] is said to divide another polynomial [math]\displaystyle{ A }[/math] when the latter can be written as
- [math]\displaystyle{ A = BC }[/math]
with C being ALSO a polynomial. This definition is similar to divisibility for integers, and the fact that [math]\displaystyle{ B }[/math] divides [math]\displaystyle{ A }[/math] is also denoted [math]\displaystyle{ B|A }[/math].
The relation between both concepts above arises when noticing the following property: [math]\displaystyle{ r }[/math] is a root of [math]\displaystyle{ P }[/math] if and only if [math]\displaystyle{ (X-r)|P }[/math]. Whereas one logical inclusion ("if") is obvious, the other ("only if") relies on a more elaborate concept, the Euclidean division of polynomials, here again strongly reminding of the Euclidean division of integers.
From this it follows that one can define prime polynomials, as polynomials that cannot be divided by any other polynomials but 1 and themselves (up to an overall constant factor) - here again the analogously with prime integers is manifest, and allows that some of the main definitions and theorems related to prime numbers and number theory have their counterpart in polynomial algebra. The most important result is the fundamental theorem of algebra, allowing for factorization of any polynomial as a product of prime ones. Worth mentioning is also the Bézout's identity in the context of polynomials. It states that two given polynomials P and Q have as greatest common divisor (GCD) a third polynomial D (D is then unique as GCD of P and Q up to a finite constant factor), if and only if there exists polynomials U and V such that
- [math]\displaystyle{ UP+VQ = D }[/math].
See also
References
- Stallings, William; : "Cryptography And Network Security: Principles and Practice", pages 121-126. Prentice Hall, 1999.
External links
- J.A. Beachy and W.D. Blair; : "Polynomials", from "Abstract algebra", 2nd edition, 1996.
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