Hilbert’s 17th Problem asks whether a real multivariate polynomial that is nonnegative on Rn can always be expressed as a sum of squares of rational functions. Emil Artin’s affirmative answer in 1927,based on the theory of real closed fields, is fundamentally non-constructive and does not provide an explicit representation algorithm. For nearly a century, this problem has inspired diverse approaches in real algebraic geometry, sum-of-squares (SOS) optimization, and effective algebra. However, a unifying theoretical framework that is inherently constructive and naturally extensible to broader function classes has remained elusive. This paper introduces, from first principles of differential algebra, the first systematic constructive and unified theory for Hilbert’s 17th Problem. The core idea is to transform the SOS representation problem into solving a system of differential-algebraic equations. Our main contributions are: (1)The precise definition of the sum-of-squares differential closure KSOS and its associated differential ideal ISOS, providing a rigorous algebraic foundation. (2) A fully computable recursive adjoining algorithm with proven formal and analytic convergence. (3) A systematic combinatorial correction theory with explicit coefficient formulas for handling high-order matching. (4) A proof that both Artin’s classical theorem and modern semidefinite programming (SDP) methods are special instances within our framework, achieving methodological unification. (5) Extensions of the theory to algebraic functions and transcendental functions with exponential weights.The resulting theory is self-contained, constructive, and explicit, offering a new paradigm that connects traditionally distinct fields and opens avenues for applications in polynomial optimization and differential inequality proving.
shifa liu (Wed,) studied this question.