KL-Minimal Solutions for Rational Interpolation

• KL-Minimal solutions are rational interpolants that minimize the sum of the degrees of the numerator and denominator, reducing algebraic complexity.
• The Extended Euclidean Algorithm generates minimal bases that track degree reductions, providing a systematic framework for identifying these optimal solutions.
• KL-minimal approaches are applied to polynomial parametrization and moving line ideals, enhancing the efficiency of modeling and interpolation in algebraic geometry.

KL-Minimal Solutions, in the context of rational interpolation and polynomial parametrization, refer to those interpolants which minimize the sum of the degrees of the numerator and denominator (the k-degree) among all possible rational solutions. This notion, formalized in the literature via the work of Kahng and further clarified in "Minimal solutions of the rational interpolation problem" (Benitez et al., 2018), is intimately connected to the structure of solutions generated by the Extended Euclidean Algorithm (EEA) and the algebraic module-theoretic framework of syzygies and bases. KL-Minimality provides a canonical measure for the "complexity" of an interpolant, especially in the algebraic-geometric paper of parametrized curves and their moving line ideals.

1. Rational Interpolation and the Notion of Minimality

Given polynomials $f(x)$ (typically the product of interpolation nodes) and $g(x)$ (e.g., a Hermite interpolant), the rational interpolation problem seeks coprime pairs $(a(x), b(x))$ of polynomials such that

$a(x) - b(x) g(x) = c(x) f(x)$

for some auxiliary $c(x)$. Solutions are typically expressed as $y(x) = a(x)/b(x)$. The set of all such pairs $(a(x), b(x))$ constitutes a module $Y$ over $K[x]$:

$$Y = \{ (a(x), b(x)) \in K[x]^2 : a(x) - b(x) g(x) \in f(x) K[x] \}$$

A minimal solution is sought, with minimality defined according either to the maximal degree of $a(x)$ or $b(x)$ (d-degree), or more generally, to the sum $k(y(x)) = \deg a(x) + \deg b(x)$—the k-degree—as in the KL-minimal definition.

2. Extended Euclidean Algorithm and Minimal Bases

The EEA, executed on $f(x)$ and $g(x)$, produces sequences of remainders $r_i(x)$, quotients $q_i(x)$, and auxiliary polynomials $s_i(x), t_i(x)$ related by recurrences:

• $r_{i+2}(x) = r_i(x) - q_{i+1}(x) r_{i+1}(x)$
• $r_i(x) = r_0(x) s_i(x) + r_1(x) t_i(x)$

Pairs $(r_i(x), s_i(x))$, $(r_{i+1}(x), s_{i+1}(x))$ serve as a basis of $Y$ for each $i$. Every possible interpolant can be represented uniquely as

$$y(x) = \frac{p(x)a_1(x) + q(x)a_2(x)}{p(x)b_1(x) + q(x)b_2(x)}$$

where $(a_1, b_1)$, $(a_2, b_2)$ form a chosen minimal basis, and $p(x), q(x) \in K[x]$ subject to non-vanishing denominator requirements at the interpolation nodes.

3. Critical Degree and KL-Minimality

A key structural insight derives from identifying a critical index $i$ in the EEA such that

$$\max\{\deg r_i(x), \deg s_i(x)\} + \max\{\deg r_{i+1}(x), \deg s_{i+1}(x)\} = n$$

where $n = \deg f(x)$ (the total data size). This split enables the minimal basis construction; denote $p_1 = \max\{\deg r_i(x), \deg s_i(x)\}$, $p_2 = \max\{\deg r_{i+1}(x), \deg s_{i+1}(x)\}$, then $p_1 + p_2 = n$. When $(a_1(x), b_1(x))$ yields coprime polynomials and the denominator does not vanish on any interpolation nodes, the unique minimal solution is $y_{\min}(x) = a_1(x)/b_1(x)$.

The minimal value of $k$ such that there exists $y(x)$ of k-degree $k$ satisfying the interpolation is called the KL-minimal value. Theorems in (Benitez et al., 2018) (particularly Theorem 5.2) detail how admissible k-degrees (and the minimum one) are determined explicitly by the EEA data.

4. Algebraic Formulation and Explicit Degree Tracking

Formulas in the EEA framework allow explicit tracking of degrees:

$$\deg r_i(x) = \deg f(x) - \sum_{j=1}^{i} \deg q_j(x)$$

The KL-minimal solution arises when the k-degree is minimized in terms of these degree sequences. If the minimal d-degree (as above) does not yield a solution (e.g., denominator vanishes at some interpolation nodes), the minimal solution is constructed from the second basis vector and a degree shift:

$$y_{\min}(x) = \frac{a_2(x) + p(x)a_1(x)}{b_2(x) + p(x)b_1(x)}$$

with $\deg p(x) = p_2 - p_1$.

5. Role in Polynomial Planar Parametrizations and Syzygies

Applications in geometric modeling, specifically in polynomial planar parametrization, necessitate the computation of u-bases or p-bases for the moving line ideal. The same critical degree found via the EEA governs minimal degree bases for such parametrizations, again linking KL-minimal solutions for interpolation to the minimal p-basis construction for parametrized curves.

The syzygetic structure revealed by the EEA thus controls both the simplicity of rational interpolants and the minimality of parametric representations, confirming the algebraic depth and practical utility of KL-minimal solutions.

6. Summary of Key Formulas and Structural Properties

Concept	Structural Formula	Context
Module of Solutions Y	$\{(a, b) : a - b g \in f K[x] \}$	Rational interpolation
KL-Minimal k-degree	$k(y(x)) = \deg a(x) + \deg b(x)$	Minimality measure
Minimal decomposition	$y(x) = [p a_1 + q a_2]/[p b_1 + q b_2]$	Any interpolant
Critical index	$$\max\{\deg r_i, \deg s_i\} + \max\{\deg r_{i+1}, \deg s_{i+1}\} = n$$	Minimal basis construction

The EEA infrastructure not only yields all rational interpolants but also encodes the degree minimality structure, culminating in the identification and efficient computation of KL-minimal solutions for rational interpolation and curve parametrization problems. This framework can be extended and adapted to related algebraic settings where interpolation and minimality are of critical interest.