Computing Constraint Sets for Differential Fields

TL;DR

This paper extends classical theorems about computable fields to differential fields, establishing key properties of constrained polynomial sets and their reducibility, advancing the understanding of computability in differential algebra.

Contribution

It adapts Kronecker's and Rabin's theorems to differential fields, demonstrating that certain aspects of these theorems remain valid in this new setting.

Findings

Two aspects of Kronecker's Theorem hold for differential fields.

One-directional reducibility property from Rabin's Theorem persists.

The paper establishes foundational results for computability in differential algebra.

Abstract

Kronecker's Theorem and Rabin's Theorem are fundamental results about computable fields F and the decidability of the set of irreducible polynomials over F. We adapt these theorems to the setting of differential fields K, with constrained pairs of differential polynomials over K assuming the role of the irreducible polynomials. We prove that two of the three basic aspects of Kronecker's Theorem remain true here, and that the reducibility in one direction (but not the other) from Rabin's Theorem also continues to hold.