Question

This exercise assumes familiarity with valuations and formal Laurent series. Let \(F\) be a field. The field \(F\left(\left(x^{-1}\right)\right)\) of formal Laurent series in \(x^{-1}\) consists of expressions of the form $$ g=\sum_{-\infty0}\). Here, \(\lfloor\cdot\rfloor\) extracts the polynomial part, so that deg \(\left(\alpha_{i}-q_{i}\right)<0\).

Short answer

The continued fraction of a Laurent series \( g \) is constructed by iteratively extracting and removing polynomial parts.

Step-by-step solution

Understanding the Setup

We start by recognizing that we are dealing with a field of formal Laurent series in the form \( g = \sum_{{-\infty < i \leq m}} g_j x^j \) with coefficients \( g_j \) in the field \( F \). The degree of such a series is defined as \( \deg(g) = \max \{ j \leq m : g_j eq 0 \} \). Our goal is to establish a continued fraction representation for such a series.

Initializing the Continued Fraction Process

We set the initial term in our process as \( \alpha_1 = g \). The continued fraction will be generated iteratively, starting with this initial Laurent series.

Extracting the Polynomial Part

For the first step in converting \( \alpha_1 \) to a continued fraction, we compute \( q_1 = \lfloor \alpha_1 \rfloor \), which is the polynomial part such that its degree is non-negative. This ensures that any terms of degree lower than 0 are discarded.

Computing the Remainder

We calculate the remainder \( \alpha_2 = \frac{1}{\alpha_1 - q_1} \). This step ensures that the resulting value has its degree below 0, allowing for the recursive continuation of the process.

Recursive Iteration

The process in Steps 3 and 4 is repeated: For every integer \( i > 1 \), compute \( q_i = \lfloor \alpha_i \rfloor \) and then \( \alpha_{i+1} = \frac{1}{\alpha_i - q_i} \). This recursive step is continued until \( \alpha_{i} - q_i \) results in terms where further computation is unnecessary or negligible.

Key concepts

Continued Fractions

Continued fractions provide a way to perform repetitive division of a number or an expression into integer parts plus a fractional remainder. This method can transform complex expressions into a more manageable series of fractions, revealing deeper insights into their structure. For formal Laurent series, continued fractions are useful for expressing a sequence of polynomial divisions leading to a fractional form.

• We commence with a given Laurent series and identify its polynomial part through extraction.
• The polynomial part forms the first term in our continued fraction sequence.
• Subsequent terms are derived recursively by considering the remainder of the series once the polynomial part is subtracted.

This recursive procedure continues, generating terms until the remainder becomes negligible, providing a detailed fractional representation.

Degree of a Polynomial

The degree of a polynomial is a fundamental concept in mathematics, referring to the highest power of the variable in the polynomial expression with a non-zero coefficient. It indicates the polynomial's complexity and potential behavior.

• For a polynomial expressed as \( p(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_0 \), the degree is \( n \).
• A higher degree generally implies more complex shapes when graphed.
• The degree of a polynomial term plays a crucial role in operations such as addition, multiplication, and determining divisibility within rational functions.

In the context of formal Laurent series, the degree provides insight into the leading term of the series, guiding the extraction of polynomial parts during the construction of continued fractions.

Field of Rational Functions

The field of rational functions is composed of quotients of polynomials, resembling a more expansive number system akin to fractions within real numbers.

• A rational function is formed as \( f(x) = \frac{p(x)}{q(x)} \), where \( p(x) \) and \( q(x) \) are polynomials and \( q(x) eq 0 \).
• This field supports all usual arithmetic operations while maintaining the structure of rationality.
• Within the field of formal Laurent series, rational functions represent a subset, showcasing simplified expressions without terms of negative degree.

Understanding rational functions can be crucial for simplifying and interpreting complex expressions, offering clarity and insight into the behavior of polynomials within larger algebraic structures.