We present a SAGBI basis detection algorithm for polynomial rings, extending a Gröbner basis detection algorithm due to Gritzmann and Sturmfels. We provide SagbiGbDetection, a software suite for Macaulay2 and Julia that identifies term orders under which an input set of polynomials forms a SAGBI or Gröbner basis. Finally, we establish complexity bounds for homogeneous SAGBI basis detection and demonstrate our implementation on novel examples.

The Parameter Continuation Theorem is the theoretical foundation for polynomial homotopy continuation, which is one of the main tools in computational algebraic geometry. In this note, we give a short proof using Gröbner bases. Our approach gives a method for computing discriminants.