Traditional computational number theory, when solving higher-degree congruence equations over finite fields Fₚ, relies heavily on the a priori determination of residue characters, causing algorithmic control flow to exhibit data-dependent logical branching. This paper establishes a fundamentally new algebro-geometric paradigm, interpreting the solution of the pure-power congruence xⁿ p as exponential evolution on a high-dimensional algebraic torus. We prove that this evolution path is uniquely determined by the structural constants of the field and is completely decoupled from the residue character of the target; the character of the target manifests itself only as an a posteriori phenomenon of whether the endpoint of evolution undergoes "dimensional degeneration" back into the base field. This framework not only provides an intrinsic geometric interpretation for the generalized Cipolla and AMM algorithms, but also reveals, at the epistemological level, the structural essence of arithmetic branching — logical branching originates from the dimensional truncation of a high-dimensional orbit space by a low-dimensional horizon. For general n-th degree equations containing intermediate terms, we point out that the transition from the non-abelian symmetric structure of a general equation to the cyclic toric structure of a pure-power equation inevitably encounters a nonlinear algebraic obstruction defined by the Tschirnhaus resolvent variety, and we thereby introduce the concept of the "Abelization threshold, " precisely delineating the applicability boundary of branch-free torus dynamics. This paper presents an affine-torus composite architecture for solving general equations, thoroughly reconstructing the algebraic roots of computational complexity.
Chuangao Ni (Fri,) studied this question.
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