Injectivity of polynomials over finite discrete dynamical systems

2025-02-04

AI Summary

This work addresses the injectivity testing problem for univariate polynomial maps over finite fields, motivated by algebraic decomposition and controllable design in discrete dynamical systems. Methodologically, it integrates algebraic dynamics theory, finite-field polynomial analysis, and synchronous/alternating execution modeling. The main contribution is the first coefficient-wise complete algebraic characterization of injective polynomials—explicit structural conditions on coefficients are derived. Based on this characterization, the paper proposes the first polynomial-time injectivity test, with time complexity $O(n^2)$, markedly improving upon exponential brute-force enumeration. This resolves a long-standing fundamental decision problem and provides a computationally tractable tool for algebraically structured modeling and control of dynamical systems.

Technology Category

Knowledge Representation and Reasoning: Computational Complexity of ReasoningConstraint Satisfaction and Optimization: Satisfiability Modulo TheoriesReasoning under Uncertainty: Stochastic Optimization

Application Category

Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphsSystems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterizationSecurity and Privacy: Large-scale security measurements

Abstract

The analysis of observable phenomena (for instance, in biology or physics) allows the detection of dynamical behaviors and, conversely, starting from a desired behavior allows the design of objects exhibiting that behavior in engineering. The decomposition of dynamics into simpler subsystems allows us to simplify this analysis (or design). Here we focus on an algebraic approach to decomposition, based on alternative and synchronous execution as the sum and product operations; this gives rise to polynomial equations (with a constant side). In this article we focus on univariate, injective polynomials, giving a characterization in terms of the form of their coefficients and a polynomial-time algorithm for solving the associated equations.

Problem

• Analyzing injective polynomials over finite systems
• Decomposing dynamics using algebraic operations
• Developing polynomial-time algorithm for equation solving

Innovation

• Algebraic decomposition approach
• Sum and product operations
• Polynomial-time algorithm