Question

This exercise discusses division with remainder when the degrees of the divisor and the quotient differ significantly. Let \(k, m \in N\) be positive. We consider univariate polynomials over an arbitrary ring (commutative, with 1, as usual). (i) Prove that division with remainder of a polynomial \(a\) of degree less than \(\mathrm{km}\) by a monic polynomial \(b\) of degree \(m\) can be done in time \((2 k+1) M(m)+O(k m)\). Hint: Partition the dividend \(a\) into blocks of size \(m\), and compute rev \({ }_{m}(b)^{-1} \bmod x^{m}\) only once. (ii) Prove that dividing a polynomial of degree \(n

Short answer

(i) \((2k + 1)M(m) + O(km)\); (ii) \((k + 3)M(m) + O(km)\). Small constant due to efficient block operations.

Step-by-step solution

Understanding Polynomial Division

Polynomial division is similar to integer division. When dividing a polynomial \( a(x) \) by another polynomial \( b(x) \), the aim is to find the quotient \( q(x) \) and remainder \( r(x) \) such that \( a(x) = q(x)b(x) + r(x) \), where the degree of \( r(x) \) is less than the degree of \( b(x) \).

Degree Constraints and Initial Setup

Given polynomials \( a \) of degree less than \( km \) and a monic polynomial \( b \) of degree \( m \), the division must be carried out efficiently. The direct approach would involve splitting \( a(x) \) into blocks of size \( m \) to standardize the process and repeating certain calculations.

Partitioning and Calculating the Inverse

By partitioning \( a(x) \) into blocks of size \( m \), initially calculate \( \text{rev}_m(b)^{-1} \mod x^m \) once. Since \( b \) is monic, we can efficiently compute this reverse polynomial due to the reduced degree, which will be critical for performing the division and understanding the repetitive calculations.

Calculate Overall Time Complexity for Part (i)

The time complexity for calculating the inverse once is \( M(m) \), and for each of the \( k \) blocks, polynomial multiplication takes \( M(m) \). Thus, for all \( k \) blocks, the time is \((2k + 1)M(m) + O(km)\). The constant within the \( O \) can be assumed to be small due to the limiting nature of block size.

Degree Analysis for Part (ii)

In the second part, the polynomial \( n < km \) is divided by another polynomial of degree \( n - m \). This involves calculating \( a(x) \) in terms of smaller units facilitated by known previous results like in exercise 8.35. Efficient division follows a similar block strategy.

Calculate Overall Time Complexity for Part (ii)

Here, the task involves fewer divisions due to the degree constraints, resulting in a time complexity of \((k + 3)M(m) + O(km)\) operations. The small constant in \( O(km) \) results from the division step efficiencies.

Key concepts

Univariate Polynomials

Univariate polynomials are mathematical expressions involving a single variable. They are like linear equations but can have variable powers higher than one, such as quadratic or cubic. Understanding univariate polynomials is important in algebra and calculus because they simplify the study of polynomial behaviors in one dimension. When we discuss the division of univariate polynomials, it’s similar to dividing numbers, looking for how many times one polynomial (the divisor) fits into another (the dividend).

In this context, we often express the division as \( a(x) = q(x)b(x) + r(x) \), where \( a(x) \) is divided by \( b(x) \), yielding quotient \( q(x) \) and remainder \( r(x) \). The remainder’s degree is less than that of the divisor, providing a clear and simple division result.

• Degree: Represented by the highest power of the variable in the polynomial.
• Polynomial Ring: A set of polynomials equipped with addition, subtraction, and multiplication operations.

Understanding how polynomials divide each other relies on these basic properties and is foundational in higher-level algebra.

Time Complexity

Time complexity is a concept that describes the amount of time an algorithm requires to complete based on the size of its input. In polynomial division, it's crucial because we want to understand how efficiently we can perform these operations.

In our exercise, time complexity arises when dividing a polynomial \( a(x) \) of degree less than \( km \) by a monic polynomial \( b(x) \) of degree \( m \). By partitioning the polynomials into segments with particular sizes, we can standardize our division process and manage computational effort.

• The time complexity to divide these polynomials involves terms like \((2k + 1)M(m) + O(km)\) for part (i) and \((k + 3)M(m) + O(km)\) for part (ii). These represent the operations count: \( M(m) \) is how long it takes to multiply polynomials of certain sizes, and \( O(km) \) captures additional operations overhead.
• Efficiency: Using the strategy of segmenting, only essential calculations are repeated, reducing unnecessary computational steps.

Time complexity helps us gauge the practicality and scalability of polynomial divisions in real-world applications.

Monic Polynomial Division

Monic polynomial division refers to the process of dividing by a polynomial whose leading coefficient (the coefficient of the highest power) is 1. This type of division is simpler since performing operations on monic polynomials eliminates the need to multiply or divide by the leading coefficient, simplifying calculations significantly.

In the exercise, the division was by a monic polynomial \( b(x) \) of a certain degree \( m \), enabling more straightforward computation. Since \( b(x) \) is monic, calculating the reverse of \( b(x) \) modulo \( x^m \) is made easier, greatly aiding efficiency in the division process.

• Reverse Polynomial: The inverse of a polynomial under modulo operations. Critical in efficient division strategies, especially in algorithms involving multiple divisions.
• Streamlined Computations: Monic properties reduce calculation overhead, focusing efforts primarily on the polynomial degrees.

Monic polynomial division supports efficient execution in computational tasks, where the ease of working with monic forms plays a significant role in algorithm efficiency.