Algebraic Curve Interpolation for Intervals via Symbolic-Numeric Computation

TL;DR

This paper introduces a novel symbolic-numeric method for algebraic curve interpolation that finds minimal degree polynomials passing through given data points or neighborhoods, optimizing the curve construction process.

Contribution

It presents a new approach to construct minimal degree algebraic curves interpolating data using optimization and symbolic computation techniques.

Findings

Effective interpolation of data with minimal degree polynomials.

Method for reconstructing integer coefficient algebraic curves.

Demonstrated efficiency through various examples.

Abstract

Algebraic curve interpolation is described by specifying the location of N points in the plane and constructing an algebraic curve of a function f that should pass through them. In this paper, we propose a novel approach to construct the algebraic curve that interpolates a set of data (points or neighborhoods). This approach aims to search the polynomial with the smallest degree interpolating the given data. Moreover, the paper also presents an efficient method to reconstruct the algebraic curve of integer coefficients with the smallest degree and the least monomials that interpolates the provided data. The problems are converted into optimization problems and are solved via Lagrange multipliers methods and symbolic computation. Various examples are presented to illustrate the proposed approaches.