Beyond Morse Theory: Trigonometric-Algebraic Functions with Weak or Absent Critical Points

This paper introduces a new class of mathematical functions—trigonometric-algebraic functions—capable of modeling energy surfaces with either very sparse or entirely absent critical points. These functions, which blend polynomial and trigonometric components, present meaningful physical and topological behaviors despite violating a central condition of Morse theory. We propose that such functions represent a new structure in differential topology and can serve as foundational models in physics, geometry, and mathematical modeling.The author acknowledges the use of OpenAI’s language model for assistance in drafting certain sections of this work.