P-schemes and Deterministic Polynomial Factoring over Finite Fields

Guo, 2017

Category: Computational Algebra

Overall Rating

2.0/5 (14/35 pts)

While the formal definition and algebraic equivalence of P-schemes offer theoretical novelty as a generalization of existing combinatorial structures, their deep entanglement with the specific algebraic context of polynomial factoring limits the immediate, actionable potential for repurposing them in unrelated domains.
Attempts to apply this highly specialized structure to problems without a clear, analogous underlying algebraic or group-theoretic foundation would likely be unproductive and redundant, as more suitable general tools exist in those fields.
However, its actionable potential for modern research is significantly limited by its dependence on the unproven GRH and the unresolved schemes conjecture for achieving polynomial-time efficiency.

The thesis introduces P-schemes as a generalization of association schemes and m-schemes, linking combinatorial structures (partitions of coset spaces) directly to group theory (posets of subgroups) and their interactions (projections, conjugations).
the core definition of P-schemes and their properties (compatibility, invariance, regularity, antisymmetry, discreteness, etc.) could be studied independently as a novel class of combinatorial objects with deep group-theoretic underpinnings.
The algebraic formulation in Appendix A (subrings of function spaces closed under certain operations) further enhances this potential, suggesting connections to areas like representation theory or functional analysis on groups...

The primary assumption grounding the algorithmic results is the Generalized Riemann Hypothesis (GRH).
Furthermore, the focus on deterministic factoring... is less urgent in practice where highly efficient randomized algorithms (like Cantor-Zassenhaus or techniques building on fast modular composition) are standard tools.
The thesis likely faded into obscurity because the path it explores—connecting deterministic factoring to complex algebraic structures derived from lifted polynomials and their Galois groups over Q, mediated by combinatorial P-schemes—is highly intricate.
Applying the specific concept of P-schemes developed here... to unrelated fields like general AI, quantum computing, or biotech would likely be a highly speculative endeavor.

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