Injective and pseudo-injective polynomial equations: From permutations to dynamical systems

TL;DR

This paper analyzes the complexity of decomposing finite discrete dynamical systems using polynomial equations, introducing pseudo-injective polynomials and providing efficient algorithms for their solutions.

Contribution

It characterizes injective and pseudo-injective polynomials over permutations and FDDSs, offering new efficient algorithms for solving related equations.

Findings

Characterization of injective polynomials over permutations and FDDSs

Efficient algorithms for solving equations involving these polynomials

Introduction of pseudo-injective polynomials with solvable equations

Abstract

We study the computational complexity of decomposing finite discrete dynamical systems (FDDSs) in terms of the semiring operations of alternative and synchronous execution, which is useful for the analysis of discrete phenomena in science and engineering. More specifically, we investigate univariate polynomials of the form $P(X) = B$, that is with a constant side, first over the subsemiring of permutations and then over general FDDSs. We find a characterization of injective polynomials $P$ and efficient algorithms for solving the associated equations. Then, we introduce the more general notion of pseudo-injective polynomial, which is based on a condition on the lengths of the limit cycles of its coefficients, and prove that the corresponding equations are also solvable efficiently. These results also apply even when permutations are encoded in an exponentially more compact way.