Question

You are to estimate the coefficient size of quotient and remainder in a pseudodivision and its cost. Let \(a, b \in \mathbb{Z}[x]\), with \(n=\operatorname{deg} a \geq m=\operatorname{deg} b>0, k=n-m\), and \(c=\operatorname{lc}(b) \in \mathbb{Z}\). Furthermore, let \(q=\Sigma_{0 \leq i \leq k} q x^{4}\) and \(r\) in \(Z\left(x \mid\right.\) be such that \(a^{*}=c^{i+1} a=q b+r\) and deg \(r0\), and assume that \(a\) has positive coefficients not smaller than \(A\), and that \(b=C x^{m}-b^{*}\) for a polynomial \(b^{*} \in Z|x|\) with deg \(b^{*}

Short answer

Step 1 gives the bounds for \(|q|\) and \(||r_i||\). In Step 2, computing costs \(O(m^2 \log^2 B)\) operations. Steps 3 and 4 confirm bounds are sharp and extend to bivariate cases.

Step-by-step solution

Understanding the Problem Setup

We have two polynomials: \(a\) of degree \(n\) and \(b\) of degree \(m\) such that \(n \geq m > 0\) and \(c\) is the leading coefficient of \(b\). We perform pseudodivision of \(a^* = c^{k+1}a\) by \(b\), yielding quotient \(q\) and remainder \(r\).

Prove Bound for \(|q|\) and \(||r_i||_{-\infty}\)

Employ the structure of pseudodivision where every step reduces the degree of the remainder. The size of \(q\) and \(r_i\) can be derived by evaluating the effect of multiplying and combining terms from \(a\) and \(b\), particularly emphasizing factors like \(c^{k-i}\) to maintain the bounds. Apply the inequality \(|q| \leq A(B+C)^{k-i} C\) and \(||r_i||_{-\infty} \leq A(B+C)^{k+1-i} C\) iteratively, acknowledging that \(||a^*||_{-} = ||c^{k+1}a||_{-}\).

Determine the Cost of Computing q and r

Recognize that the computation cost involves multiplying and adding polynomials with bounded coefficients. The pseudodivision process iteratively reduces degrees using steps like subtraction and multiplication, similar to the polynomial division process but adapted for integer coefficients. Given \(A \leq B\), use complexity bounds to conclude that this takes \(O(m^2 \log^2 B)\) word operations.

Show Lower Bound for q_i and r

For \(q_i\), analyze the polynomial structure, considering it results from consistent subtraction of \(b\)'s multiples scaled up by \(c\), leading to \(q_i \geq A(B+C)^{k-i} C\). Similarly, since \(a\)'s coefficients are positive and larger or equal to \(A\), the accumulated effect through division yields \(||r||_{-}\geq A(B+C)^{k+1}\). These support the sharpness of the bounds from part (i).

Analogous Statements for Bivariate Polynomials

In a similar fashion, one must consider each variable independently in polynomial division. The bivariate case demands addressing complexity separately for each variable, respecting the interaction between degrees in \(x\) and \(y\), leading to cost bounds and quotient-remainder bound structures that are analogous to the univariate case. The complexity generally scales with the product of the degrees of \(b\) in both variables.

Key concepts

Pseudodivision

Pseudodivision is an algorithm used in polynomial division when working with integer coefficients, where traditional polynomial division might result in fractions. To avoid fractions, pseudodivision scales the dividend by a power of the leading coefficient of the divisor. This ensures that all intermediate coefficients remain integers. When a polynomial \(a\) is pseudodivided by \(b\), it involves multiplying \(a\) by the leading coefficient \(c\) of \(b\) raised to a certain power. This adjustment helps maintain integer coefficients throughout the division.

• This process yields a quotient \(q\) and a remainder \(r\) without fractions.
• At each stage, the degree of the remainder decreases, promoting convergence towards the actual quotient and remainder.

By pseudodividing \(a^* = c^{k+1}a\), we gain a scaled version of the original quotient and remainder, enhancing the computational integrity of the result.

Polynomial Bounds

When performing operations like pseudodivision, it's crucial to establish bounds for the coefficients involved. For example, in our context, estimating the maximum size of the coefficients in both the quotient \(q\) and the remainder \(r\) is essential to ensure that computation remains feasible.

• The bound \(|q| \leq A(B+C)^{k-i} C\) is derived by estimating the effect of scaling and multiplying terms from \(a\) and \(b\).
• The size estimation for \(r_i\) ensures no unexpected growth in magnitude beyond \(A(B+C)^{k+1-i} C\).

These bounds are vital as they allow us to predict the result's size, ensuring that our calculations won't exceed practical limits during computation. Understanding these limits is critical, especially in controlled algorithms like pseudodivision.

Computational Complexity

The computational complexity of pseudodivision directly impacts the efficiency of polynomial operations. The process involves reducing polynomial degree iteratively, similar to traditional polynomial division, yet adapted for integers to enhance accuracy.

• For polynomials where \(A \leq B\), the complexity is shown to be \(O(m^2 \log^2 B)\), which indicates the relationship between the polynomial degree and coefficient size.
• This bound arises because pseudodivision incorporates multiple operations, including multiplications and additions, which are managed within the given cost constraint.

This efficient method ensures that even with high-degree or high-coefficient polynomials, the operations remain manageable, making pseudodivision a preferred choice in computational mathematics involving integers.

Bivariate Polynomials

Bivariate polynomials involve two different variables, often denoted as \(x\) and \(y\). When conducting operations such as pseudodivision on bivariate polynomials, it's essential to consider how each variable interacts and affects the overall complexity.

• The division must assess the degree in each variable independently, often leading to more intricate calculations than univariate cases.
• Cost bounds and coefficient estimations must account for the simultaneous impact of both variables, potentially resulting in increased complexity.
• The complexity in handling bivariate polynomials can scale with the product of the degrees of \(b\) regarding both variables, hence demanding careful execution.

Understanding the nuances of bivariate polynomial operations ensures more accurate computations in multivariate scenarios, making them indispensable in advanced algebraic methodologies.