Sums of Even Powers in Rational Function Fields Over The Field of Real Numbers: The Case of Fourth Powers

Abstract

This study focus on sums of even powers in field of rational function over the field of real num bers; specifically sums of fourth powers. The main idea of the study extended from Hilbert’s seventeenth problem. Hilbert’s Seventeenth problem state that “Is it possible to express a mul tivariate polynomial that takes only non-negative values over the reals as a sum of squares of rational functions”?. In 1927, Artin demonstrated that a real polynomial, which is posi tive semidefinite, can indeed be represented as a sum of squares of rational functions, thereby resolving Hilbert’s 17th problem. The natural extension of Hilbert’s seventeenth problem to in clude sums of even powers has been explored by several mathematicians during the mid-20th century. In 1981, Becker established conditions that allows non-negative rational functions to be expressed as sums of 2m-th powers, where m is a positive integer and for a real function f ield F in one variable over a real closed field, he established that P4(F) ≤ 36. In 1996, Choi, Lam and Reznick showed that for any formally real field R, any sums of fourth power in R(x) can be represented as at most 6 fourth powers of rational functions using geometric argument. This research aims to investigate the criteria for expressing rational functions or polynomials as sums of fourth powers and to offer an alternative proof for the effective upper bound of such sums.