Question

Let \(F\) be a field of characteristic zero and \(a, b \in F[x]\) nonzero and coprime. (i) Let \(\gamma \in F\) and \(p \in F[x]\) be an irreducible factor of \(\operatorname{gcd}\left(b, a-\gamma b^{\prime}\right)\). Prove that \(p^2 \nmid b\), and conclude that the gcd is squarefree. (ii) Show that \(\operatorname{gcd}\left(b, a-\gamma_1 b^{\prime}\right)\) and \(\operatorname{gcd}\left(b, a-\gamma_2 b^{\prime}\right)\) are coprime if \(\gamma_1, \gamma_2 \in F\) are distinct. (iii) Consider the following variant of Algorithm 22.14. ALGORITHM \(22.22\) Almkvist \& Zeilberger's multiple of integration denominator. Input: Relative prime polynomials \(a, b \in F[x]\) with \(b \neq 0\) monic. Output: A monic polynomial \(V \in F[x]\) such that for any coprime \(u, v \in F[x]\), equation (5) implies that \(v\) divides \(V\). 1\. \(R \longleftarrow \operatorname{res}_x\left(b, a-y b^{\prime}\right), \quad d \longleftarrow \max \\{i \in \mathbb{N}: i=0\) or \(R(i)=0\\}\) if \(d=0\) then return 1 2\. \(a_0 \longleftarrow a, \quad b_0 \longleftarrow b\) for \(i=1, \ldots, d\) do $$ H_i \longleftarrow \operatorname{god}\left(b_{i-1}, a_{i-1}-b_{i-1}^{\prime}\right), \quad a_i \longleftarrow \frac{a_{i-1}-b_{i-1}^{\prime}}{H_i}, \quad b_i \longleftarrow \frac{b_{i-1}}{H_i} $$ 3\. return \(H_1 H_2^2 \ldots H_d^d\) Show that \(H_i=\operatorname{gcd}\left(b, a-i b^{\prime}\right)\) for all \(i\) and conclude that the algorithm returns the same result as Algorithm 22.14. (iv) Trace Algorithm \(22.22\) on the input from Example 22.16.

Short answer

(i) The gcd \(p^2 \nmid b\) is squarefree. (ii) Distinct \(\gamma\) lead to coprime gcds. (iii) Each \(H_i\) is consistent, matching Algorithm 22.14.

Step-by-step solution

Understanding coprimeness and gcd

Since \(a\) and \(b\) are coprime, there are no common factors between them. When we compute \(\operatorname{gcd}(b, a - \gamma b')\), any factor \(p\) that divides this gcd implies some interaction between \(b\) and \(a - \gamma b'\).

Show \(p^2 \nmid b\)

Suppose \(p^2 \mid b\). Then \(p \mid b\) implies \(p \mid a - \gamma b'\). Since \(b = p\cdot q\), differentiating gives \(b' = p'q + pq'\). Hence \(a - \gamma b'\) becomes part of \(p\), and \(p\) dividing both contradicts the fact that \(a\) and \(b\) are coprime, i.e., \(p^2 mid b\).

Conclude gcd is squarefree

From Step 2, since \(p^2 mid b\) and \(a\), \(b\) are coprime, the irreducible gcd \(\operatorname{gcd}(b, a - \gamma b')\) must be squarefree. Hence, no squared factors can exist.

Distinct \(\gamma\) values lead to coprime results

Given \(\gamma_1\) and \(\gamma_2\) are distinct, consider the polynomials \(a - \gamma_1 b'\) and \(a - \gamma_2 b'\). If there existed a common factor between \(\operatorname{gcd}(b, a - \gamma_1 b')\) and \(\operatorname{gcd}(b, a - \gamma_2 b')\), it would contradict the distinct nature of their gcd definitions. Thus, they are coprime.

Analysing Algorithm 22.22

The algorithm uses a step-by-step computation since \(H_i = \operatorname{gcd}(b_{i-1}, a_{i-1} - i b'_{i-1})\) as stated. It aligns with the definition for the gcd being used, ensuring that \(H_1 H_2^2 \ldots H_d^d\) is equivalent to the other algorithm, as polynomial differentiation leads to consistent result forms.

Key concepts

Polynomial GCD

The greatest common divisor (GCD) of two polynomials is a fundamental concept in computer algebra. It signifies the largest polynomial that divides two given polynomials without leaving a remainder. When dealing with polynomials over a field, the GCD can be computed efficiently using algorithms similar to the Euclidean algorithm used for integers.

In the context of the exercise, considering the polynomials \(a\) and \(b\), which are coprime (i.e., their GCD is 1), helps demonstrate how a new polynomial, derived as \(a - \gamma b'\), interacts with \(b\). Calculating the GCD of \(b\) and \(a - \gamma b'\) identifies if there are any new common factors introduced, which is crucial in further understanding the nature of these polynomials, such as their irreducibility and squareness.

Understanding how the GCD is impacted by differentiable transformation \((b')\) and scalar multiplication \((\gamma)\) is critical in algorithm analysis, particularly, it shows how polynomials preserve or lose their factorizations.

Irreducible Factors

An irreducible factor is a polynomial that cannot be factored further into nontrivial divisors over its field. It's the polynomial equivalent of a prime number in integer arithmetic. In the exercise, the focus on irreducible factors stems from understanding how the GCD interacts with such factors.

The key takeaway from irreducible factors in this context is the implication that if \(p\) is an irreducible factor of some polynomial, then \(p^2\) does not divide that polynomial to maintain its irreducibility. The reasoning behind this lies in if \(p^2\) divides the polynomial, \(p\) itself becomes reducible within that context, which contradicts its definition.

This property's significance is paramount when checking whether a polynomial GCD is squarefree, as seen in earlier steps of the exercise. Identifying irreducibility helps ensure computations are precise, especially when polynomials are manipulated in algorithmic processes.

Squarefree Polynomials

A squarefree polynomial is one that does not have any squared factors; that is, no irreducible polynomial divides it more than once. The exercise's focus on squarefree polynomials is instrumental because such polynomials simplify computations significantly.

When a polynomial's GCD is squarefree, it implies that while there may be common factors in the polynomial expression, none will be repeated, or duplicated.

• This property ensures distinct divisibility, making it particularly useful when designing algorithms.
• A squarefree polynomial eases computational complexity, as it avoids redundant factors, enhancing efficiency in polynomial division-based operations.

Detecting squarefree characteristics in a polynomial GCD provides confidence in the uniqueness and simplicity of any solutions generated from derived expressions.

Algorithm Analysis

Algorithm analysis in computer algebra entails examining the efficiency and effectiveness of algorithms used to solve polynomial equations. For the exercise, this involves analyzing the variant Algorithm 22.22.

This algorithm intends to output a monic polynomial, which maintains coprimeness conditions across several steps:

• The computation of successive GCDs \(H_i = \operatorname{gcd}(b, a - i b')\) at each step establishes a sequence relevant to divisibility properties.
• The final output being the product \(H_1 H_2^2 \ldots H_d^d\) ensures that the resulting monic polynomial accounts for efficiency, with complexity designed to minimize unnecessary calculations, optimizing performance.

By tracing this algorithm, students can visualize how theoretical concepts translate into practical applications, highlighting intersections between polynomial manipulation and algorithmic efficiency.