Question

Given are a positive integer \(n \in \mathbb{N}\), two polynomials \(a=\sum_{0 \leq i

Short answer

(i) The upper bound for \(|c_i|\) is \(nB^2\). (ii) Multiply modulo small primes, then combine using CRT. (iii) Compute each step modulo primes (e.g., using 5, 7) numerically.

Step-by-step solution

Analyze Polynomial Coefficient Products

When multiplying two polynomials \(a\) and \(b\), each coefficient \(c_i\) of the product polynomial \(c = ab\) is the sum of products \(a_jb_{i-j}\) for suitable \(j\). Since \(|a_j| \leq B\) and \(|b_{i-j}| \leq B\), \(|a_j b_{i-j}| \leq B^2\). For each \(c_i\), there are at most \(n\) such terms (since \(j\) can take from 0 to \(n-1\)), so the maximum \(|c_i|\) is \(nB^2\).

Determine Upper Bound on |c_i|

Thus, a tight common upper bound for \(\left|c_i\right|\) is \(nB^2\). This result assumes the worst-case scenario where all \(n\) terms contributing to \(c_i\) are equal to the product \(B^2\).

Overview of Small Primes Modular Approach

The small primes modular approach involves computing the polynomial product modulo small primes. Choose small primes \(p_1, p_2, \ldots, p_k\) such that their product exceeds the possible largest coefficient value \(\leq nB^2\). Compute \(c(x) \equiv a(x)b(x) \pmod{p_j}\) for each prime, and then reconstruct \(c(x)\) using the Chinese Remainder Theorem to find coefficients in \(\mathbb{Z}\).

Trace Algorithm with Given Polynomials

Given \(a=987x^3+654x^2+321x\) and \(b=-753x^3-333x^2-202x+815\), first calculate the coefficients of the product polynomial separately modulo small primes (e.g., choose primes 2, 3, 5 for simplicity). For each \(p_j\), compute \(a(x) \mod p_j\) and \(b(x) \mod p_j\), multiply them to get \(c(x) \mod p_j\), and then use the Chinese Remainder Theorem to reconstruct \(c(x)\). Working through a few of these computations as an illustration is necessary for full tracing.

Execute a Specific Modular Computation (Example)

Using prime \(p=5\), calculate the polynomials modulo 5: \(a(x) \equiv 2x^3 + 4x^2 + x \mod 5\) and \(b(x) \equiv 2x^3 + 2x^2 + 3 \mod 5\) from the coefficient reductions. Multiply \(a\) and \(b\) modulo 5 which results in a polynomial equation modulo 5, e.g., \(c(x) = a(x)b(x) \mod 5\). Repeat the process for other chosen primes and combine results using the Chinese Remainder Theorem.

Key concepts

Chinese Remainder Theorem

The Chinese Remainder Theorem (CRT) is a critical tool in computational mathematics, particularly in polynomial multiplication using the modular approach. The theorem states that if you have several congruences to solve, each modulo different pairwise coprime numbers, there exists a unique solution modulo the product of these numbers. This theorem is most beneficial when computing polynomial products modulo several small primes, bridging them into a final solution.

• It allows us to reduce large computations into simpler problems.
• Individual modular computations can be performed independently and are generally faster.
• The theorem also ensures that reconstruction of values is unambiguous and efficient.
  Considered collectively, these make CRT an ideal choice for reducing potential computational difficulties in polynomial multiplication.

When reconstructing the final polynomial coefficients, CRT helps to combine each separate result into a single polynomial expression with integer coefficients.

Modular Arithmetic

Modular arithmetic is the foundation of the small primes modular approach in polynomial multiplication. In essence, it deals with integers within a limiting boundary set by a modulus; once surpassed, numbers "wrap around" to start from zero. When multiplying polynomials, performing operations under modulus simplifies calculations, since each coefficient cannot exceed the modulus.

• This method helps limit the size of individual numbers, making them easier and faster to handle during calculations.
• It is especially useful in reducing overflow errors and ensuring computations are more manageable.
• By choosing appropriate small primes, you guide the process to stay efficient and accurate.
  

Overall, using modular arithmetic transforms the problem into more feasible units that can be recombined into a final answer using additional methods like CRT.

Worst-case Analysis of Coefficients

Understanding the worst-case scenario in polynomial multiplication is key to effective algorithm design. In the given problem, each coefficient of the resulting polynomial has a maximum potential value, calculated during a worst-case analysis. This aspect assesses what happens when each contributing term reaches its largest possible value.

• For our polynomials, this maximum value is defined as \(nB^2\), where \(n\) represents the number of terms and \(B\) is the bounded coefficient value.
• Considering this worst-case expands upon how one might ensure the accuracy and robustness of computational methods.
• These calculations guide the choice of modulus and primes in the modular method, ensuring reliability focused on worst-case limits.
  

The purpose is to anticipate the extent of computation needs and ensure algorithmic methods can manage most demanding cases mathematically.

Algorithmic Efficiency

Algorithmic efficiency in polynomial multiplication involves minimizing time and space complexity during calculations. Leveraging the modular approach paired with CRT optimizes efficiency by dividing tasks into smaller, parallelizable units. Given that polynomial calculations can be extensive, efficiency translates into reduced computation times and demands.

• Applying smaller modular "chunks" allows rapid calculations with decreased resource use.
• Efficiency is further enhanced by the independent nature of calculations per prime, allowing easy parallel processing.
• It supports achieving well-rounded results without needing excessively large computational resources.
  

Understanding and implementing efficient algorithms balance speed and accuracy, making feasible what might otherwise be computationally unachievable with direct methods.