Technology Category
Search and Optimization: Mixed Discrete/Continuous SearchConstraint Satisfaction and Optimization: Distributed CSP/OptimizationMultiagent Systems: Distributed Problem Solving
Injective and pseudo-injective polynomial equations: From permutations to dynamical systems
🤖 AI Summary
This work addresses the challenging problem of solving polynomial equations involving division and $k$-th roots in connected finite discrete-time dynamical systems (FDDS). The authors propose an efficient algorithm based on an “unrolled” representation, leveraging a novel backward infinite forest-of-trees model. For the first time, this approach yields polynomial-time algorithms for division and $k$-th root operations in connected FDDS, enabling effective solution of equations of the form $AX^k = B$ and their generalizations. By exploiting the inherent algebraic structure of FDDS together with tree-forest modeling techniques, the method not only achieves efficient solvability for specific equation classes but also establishes a foundational theoretical and algorithmic framework for investigating the solvability of general polynomial equations in FDDS.
Technology Category
Application Category
📝 Abstract
Finite discrete-time dynamical systems (FDDS) model phenomena that evolve deterministically in discrete time. It is possible to define sum and product operations on these systems (disjoint union and direct product, respectively), giving a commutative semiring. This algebraic structure led to several works employing polynomial equations to model hypotheses on phenomena modelled using FDDS. To solve these equations, algorithms for performing division and computing $k$-th roots are needed. In this paper, we propose two polynomial algorithms for these tasks, under the condition that the result is a connected FDDS. These algorithms exploit the notion of unroll of a FDDS, an alternative representation based on a forest of infinite trees constructed by computing the transition function of the system backwards. This ultimately leads to an efficient solution to equations of the type $AX^k=B$ for connected $X$ and some generalisations. These results are some of the important final steps for solving more general polynomial equations on FDDS.
Problem
Research questions and friction points this paper is trying to address.
finite discrete-time dynamical systems
polynomial equations
connected FDDS
k-th roots
division algorithms
Innovation
Methods, ideas, or system contributions that make the work stand out.
finite discrete-time dynamical systems
polynomial equations
unroll representation
k-th root algorithm
connected FDDS
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