Question

Let \(F\) be a field, \(m, n \in \mathbb{N}\) with \(m

Short answer

The subresultant is \(\sigma_h = \operatorname{det} A_h\); constant coefficient of \(t\) is \(\operatorname{det} B_h / \operatorname{det} A_h\). Padé approximant exists if \(\operatorname{det} B_h \neq 0\).

Step-by-step solution

Define the Subresultant Polynomial

The subresultant polynomial of two polynomials is related to the construction of a matrix from their coefficients. Specifically, the subresultant matrix associated with two polynomials involves the coefficients of each polynomial up to the degree specified by the subresultant order. In this case, we need to find the subresultant of the polynomials \(x^n\) and \(f\). Since \(x^n\) has only one non-zero coefficient, many entries of \(A_h\) will simplify quickly.

Construct the Matrix \(A_h\)

The matrix \(A_h\) is constructed with entries \((f_{h-i+j})\) for \(0 \leq i, j < n-h\). This involves creating a matrix of size \((n-h)\times(n-h)\). The indices \(i\) and \(j\) define the rows and columns, respectively, where each entry is determined by shifting the coefficients of \(f(x)\). Note that \(f(x)\) has degree \(m\).

Calculate the Determinant of \(A_h\) for Subresultant Value

The subresultant \(\sigma_h\) of the two polynomials is given to be the determinant of matrix \(A_h\), i.e., \(\sigma_h = \text{det} A_h\). This follows because the determinant essentially captures the resultant of the polynomial \(f(x)\) with \(x^n\) under certain shifts in coefficients, thus matching the properties of the subresultant.

Implement Cramer's Rule

For part (ii), given that \(\sigma_h eq 0\) implies a non-zero determinant, we use Cramer's rule for solving systems of linear equations applied to polynomial coefficients. This involves using the determinant of \(B_h\) to find the specific coefficient of interest in the polynomial remainder when performing the Euclidean algorithm on \(x^n\) and \(f(x)\).

Prove the Constant Coefficient Relation

Given the hint and Cramer's rule, the constant coefficient of the polynomial \(t\) (found in the Euclidean algorithm relation for \(x^n\) and \(f\)) is given by \(\text{det} B_h / \text{det} A_h\). This follows from the linear dependence of the resultant (based on subresultants) related to these matrix determinants.

Evaluate Existence of Padé Approximant

Finally, for part (iii), the existence of an \((k, n-k)\)-Padé approximant is determined by checking the condition that \(\text{det} B_h eq 0\) given \(\sigma_h eq 0\). This relates directly to the condition that such an approximant exists if and only if the matrix formed by cutting out the last rows and columns still retains a non-zero determinant.

Key concepts

Cramer's Rule

Cramer's Rule is an efficient mathematical theorem used to solve linear systems of equations with as many equations as unknowns, assuming the determinant of the system's coefficient matrix is non-zero. It's particularly handy because it provides an explicit formula for each variable in the system using determinants.

To employ Cramer's Rule, follow these steps:

• First, find the determinant of the matrix formed by the system's coefficients, called the main determinant.
• Next, to solve for a given variable, substitute the respective column of the coefficient matrix with the column vector of constants from the equations.
• Then, calculate the determinant of this new matrix.
• The value of the variable is the ratio of this new determinant to the main determinant.

In the context of polynomial exercises, Cramer's Rule can be used to find specific coefficients in the remainder when dividing one polynomial by another. Here, it aids in obtaining the constant coefficient of the polynomial remained in a division, linking directly to the subresultant matrices' determinants.

Matrix Determinant

The matrix determinant is a crucial concept in linear algebra representing the scaling factor by which a transformation described by the matrix changes areas or volumes in space. When dealing with square matrices, its determinant is a single scalar value.

In polynomial theory, determinants help in determining the subresultant of two polynomials, a specific form of the resultant of two polynomials when one has been shifted by a certain degree:

• The determinant of a matrix corresponding to polynomial coefficients, like the defined matrix \(A_h\), encodes vital information about the roots and factors of polynomials under consideration.
• These determinants can indicate properties like solvability or uniqueness of solutions in linear transformations.
• For matrices derived from two polynomials, like \(x^n\) and \(f\), the determinant of \(A_h\) is particularly telling of the existence of specific polynomial approximations and their relationships.
  
  

When computing determinants:

• Utilize methods like Laplace expansion or Gaussian elimination for efficiency, especially for larger matrices.
• In polynomials, the determinant establishes connections to subresultants, essentially narrowing down polynomial roots and aiding in computations like Padé approximations.

Padé Approximation

Padé Approximations provide an excellent method for approximating functions with rational functions, which are quotients of two polynomials. This is particularly beneficial in numerical methods, where it helps in approximating complex functions with simpler forms.

These approximations are recognized for capturing the function's essence better than Taylor series at times, as they can handle singularities in a more comprehensive manner:

• A \(k, n-k\)-Padé approximant represents the closest approximation of a given polynomial, \(f\), where k is the degree of the numerator polynomial, and \(n-k\) is the degree of the denominator polynomial.
• In specific exercises involving subresultants, the Padé approximation hinges on the property that \(\text{det} B_h eq 0\) given \(\sigma_h eq 0\), ensuring the corresponding rational function truly approximates the target function effectively.
• This is contingent upon having a non-zero submatrix determinant, ensuring the accuracy and validity of the approximation.

Understanding and applying Padé Approximations can significantly enhance proficiency in managing polynomial functions and understanding their behavior over complex domains. It’s a powerful technique, well integrated with determinants and subresultant theory, ensuring computational efficiency and robustness in function approximation.